# A local phase space stochastic quantization?

Can<EMAIL_ADDRESS>
###### Abstract

I examine whether Nelson’s stochastic formulation of Schrödinger equation
could be derived from a phase space process through a colored noise smoothing.
If this conjecture is true, it would yield a local stochastic hidden variable
theory. I discuss how this does not necessarily contradict Bell type theorems
as general local stochastic theories can violate local causality assumptions.
I also discuss the generalization to quantization of fields and speculate
about the gravitational origins of noise.

###### Contents

  1. 1 Introduction
  2. 2 Single particle
    1. 2.1 Nelson’s stochastic mechanics for a single particle
    2. 2.2 Colored noise smoothed quantum process
    3. 2.3 Calculation of stochastic acceleration for a colored noise process
    4. 2.4 A phase space derivation?
    5. 2.5 Conjecture 1 implies Conjecture 2
  3. 3 Multiple particles
    1. 3.1 Nelson’s mechanics for multiple particles
    2. 3.2 A phase space derivation for multiple particles?
    3. 3.3 Conjecture 3 implies Conjecture 4
  4. 4 Locality and local causality
    1. 4.1 Nonlocality of Nelson’s theory vs locality of phase space model
    2. 4.2 Multi-time measurements in stochastic mechanics
    3. 4.3 Local causality
  5. 5 Fields
    1. 5.1 Potential physical sources of noise
    2. 5.2 Stochastic quantization of the free scalar field
    3. 5.3 Gravitational origins of noise

##  1 Introduction

Ever since the advent of quantum theory, the question of whether there exists
a local deterministic or stochastic theory underlying quantum mechanics has
been of considerable interest. One important direction was provided by
Nelson’s seminal work on the reformulation of quantum mechanics in terms of
stochastic mechanics of particles [1, 2, 3]. Stochastic mechanics is an exact
reformulation of quantum theory like the path integral formulation. However,
when applied to multiple particles and fields, it is manifestly non-local.
Nelson suggested the important question [3]: Can one give a local model which
would reduce to stochastic mechanics in an appropriate limit?. This is the
central question, I am concerned with in this note. Although, it is folklore
to believe that it is impossible, I argue following Nelson that Bell type
theorems do not actually rule out local stochastic theories which in general
can violate local causality assumptions. I outline the construction of a
simple local theory and formulate conjectures which would enable such a theory
to exist first for the single particle case, then for multiple particles and
finally for the prototypical example of the scalar field. Of course, existence
of such a theory does not mean that the universe obeys it. Assuming that the
theory is physical, I will also speculate about potential gravitational
origins of noise. In the absence of evidence for the quantization of the
gravitational field, this could provide a consistent view of the universe
where quantum theory arises out of gravitational interactions for all the
fields in the universe. This is similar to that Brownian motion of a single
particle arises from the microscopic Newtonian dynamics of a gas.

To see the main difficulty, consider the Madelung formulation of quantum
theory:

$\frac{\partial\rho}{\partial t}=-\frac{\partial}{\partial
x}(\rho\frac{1}{m}\frac{\partial S}{\partial x})$ (1.1)

$\frac{\partial S}{\partial t}=-\frac{1}{2m}(\frac{\partial S}{\partial
x})^{2}-U(x)+\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}\frac{\partial^{2}}{\partial
x^{2}}\sqrt{\rho}$ (1.2)

where $\rho(x,t)$ is the probability of finding the particle at $(x,t)$ and
$S(x,t)$ is the phase of the wave function. The second equation would be the
classical Hamilton-Jacobi equation if the last term (quantum potential) was
absent, and the equations would describe an ensemble of classical particles.
Thus, a derivation of Schrödinger equation must somehow generate the quantum
potential term either implicitly or explicitly. Nelson’s stochastic mechanics
in terms of particles trajectories (see section 2.1) provides an implicit
understanding of this term in terms of averaged stochastic derivatives. If
there are multiple particles, the quantum potential term in general depends on
all the positions of the particles. A stochastic unraveling such as Nelson’s
theory then becomes non-local. The central question is equivalent to asking
whether one can induce the quantum potential term while retaining locality.
Nelson argued that Markovianity of his theory might be the main obstacle
against formulating a local underlying theory and a non-Markovian theory is
not necessarily non-local in [3]. I will propose a way to circumvent this
problem through replacing the white noise assumption in Nelson’s theory with
colored noise. One important objection against the existence of a such a local
theory comes from believes surrounding Bell-type theorems or inequalities.
These theorems assumes two kinds of locality: the usual locality in the sense
of relativity and local causality. Local causality assumptions do not have to
be satisfied by general local stochastic theories. I will discuss these issues
in section 4.

The outline is as follows. In section 2, I give an overview of Nelson’s theory
for a single particle then propose a phase space formulation based on colored
noise. I formulate conjectures related to the existence of such a theory. In
section 3, I generalize the arguments given for the single particle case to
the multiple particle case. In section 4, I elaborate on the issues of
locality and local causality. In section 5, I generalize the ideas to the case
of the scalar field and speculate about the gravitational origin of noise.

##  2 Single particle

###  2.1 Nelson’s stochastic mechanics for a single particle

I give a brief review of Nelson’s stochastic formulation of non-relativistic
quantum mechanics in one dimension. For more details see Nelson’s original
paper[1], his two books[2, 3] and Guerra’s review[4]. Consider the Schrödinger
equation:

$i\hbar\frac{\partial\psi(x,t)}{\partial
t}=(-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+U(x))\psi(x,t).$
(2.1)

Putting $\psi(x,t)=\sqrt{\rho(x,t)}e^{\frac{i}{\hbar}S(x,t)}$ we get the
Madelung equations:

$\frac{\partial\rho}{\partial t}=-\frac{\partial}{\partial
x}(\rho\frac{1}{m}\frac{\partial S}{\partial x})$ (2.2)

$\frac{\partial S}{\partial t}=-\frac{1}{2m}(\frac{\partial S}{\partial
x})^{2}-U(x)+\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}\frac{\partial^{2}}{\partial
x^{2}}\sqrt{\rho}$ (2.3)

where $\rho(x,t)$ is the probability of finding the particle at $(x,t)$ and
$S(x,t)$ is the phase of the wave function. We recognize first of the
equations as the continuity equation with velocity $\frac{1}{m}\frac{\partial
S}{\partial x}$. The second of the equations apart from the last term (quantum
potential) on the right hand side is the Hamilton-Jacobi equation. Thus if
$\hbar=0$, we have the classical ensemble of particles. The Newton’s equations
of motion are then the equations that characteristic curves obey corresponding
to this set of Madelung partial differential equations. Since the quantum
potential term depends on the probability $\rho(x,t)$, giving deterministic
characteristics seems not possible. However as Nelson proved[1, 2, 3], it is
possible to give a Markovian stochastic process associated to the solution of
Madelung equations in position space. We start by assuming that a particle
obeys the following stochastic differential equation:

$dx(t)=b(x(t),t)dt+\sqrt{\frac{\hbar}{m}}dW(t)$ (2.4)

where $b(x(t),t)$ is a general function and $dW$ is the Wiener process. We
will always interpret the stochastic differential equations in this paper in
Itô sense. We call this as Nelson’s first postulate. The diffusion equation
associated to this is[5, 6]

$\frac{\partial\rho(x,t)}{\partial t}=-\frac{\partial}{\partial
x}(b(x,t)\rho(x,t))+\frac{\hbar}{2m}\frac{\partial^{2}}{\partial
x^{2}}\rho(x,t)$ (2.5)

where $\rho(x,t)$ is the probability of finding the particle at $x$ at time
$t$. In order to match with the continuity equation we define

$\frac{\partial}{\partial
x}S(x,t)=m(b(x,t)-\frac{\hbar}{2m}\frac{\partial}{\partial x}\log\rho(x,t))$
(2.6)

where we assumed that $\rho(x,t)$ is nowhere zero. For a discussion of what
happens at zeros see [3] and references therein. Indeed, the probability of
the particle arriving at places where $\rho(x,t)=0$ is zero since $b(x,t)$ is
singular at zeros and acts repulsively on the stochastic particle trajectory
pushing the particle away from the zeros. We want $S(x,t)$ just defined in
this way to satisfy the quantum Hamilton-Jacobi equation. We could postulate
it as a partial differential equation but Nelson found a way to write this
solely in terms of the stochastic particle trajectory. The quantum Hamilton-
Jacobi equation can be shown to be equivalent to the following equation:

$\frac{1}{2}(D_{+}D_{-}+D_{-}D_{+})x(t)=-\frac{1}{m}\frac{\partial
U(x)}{\partial x}|_{x(t)}$ (2.7)

where $D_{+}$ and $D_{-}$ are forward and backward derivatives which will be
defined below, the right hand side is the classical acceleration of the
particle evaluated on the stochastic trajectory and the left hand side is the
time-symmetric stochastic acceleration. This is the stochastic analogue of
Newton’s second law. Thus we call this as Newton-Nelson law or Nelson’s second
law. The forward and backward derivatives are defined to be

$D_{+}x(t)=\lim_{\Delta t\to 0^{+}}E[\frac{x(t+\Delta t)-x(t)}{\Delta
t}|x(t)]$ (2.8)

$D_{-}x(t)=\lim_{\Delta t\to 0^{+}}E[\frac{x(t)-x(t-\Delta t)}{\Delta
t}|x(t)]$ (2.9)

where $E[f|x(t)]$ denotes the expectation of $f$ conditioned on $x(t)$. For
any function $F(x,t)$ we can write its forward and backward derivatives
explicitly as follows

$(D_{+}F)(x,t)=\frac{\partial}{\partial
t}F(x,t)+b(x,t)\frac{\partial}{\partial
x}F(x,t)+\frac{\hbar}{2m}\frac{\partial^{2}}{\partial x^{2}}F(x,t)$ (2.10)

$(D_{-}F)(x,t)=\frac{\partial}{\partial
t}F(x,t)+(b(x,t)-\frac{\hbar}{m}\frac{\partial}{\partial
x}\log\rho(x,t))\frac{\partial}{\partial
x}F(x,t)-\frac{\hbar}{2m}\frac{\partial^{2}}{\partial x^{2}}F(x,t).$ (2.11)

The derivation of the formula for $D_{+}$ is straightforward but the
calculation of $D_{-}$ is subtler[2, 3, 4]. Using these formulas it is
straightforward to show that the Newton-Nelson law is equivalent to the $x$
derivative of the second Madelung equation (eq.2.3). It has been shown that
for each solution of the Schrödinger equation there is an associated
stochastic process satisfying Nelson’s postulates and if Nelson’s postulates
are satisfied that one can construct a wave function which satisfies the
Schrödinger equation with its absolute square the probability density of the
position of particle. The stochastic formulation can be generalized to
particles propagating in higher dimensions, multiple particles, fields and
particles with spin[3, 4].

###  2.2 Colored noise smoothed quantum process

Let $A_{\beta}$ be the colored noise defined by the equation

$dA_{\beta}=-\beta A_{\beta}dt+\beta dW$ (2.12)

where $dW$ is the Wiener process. Heuristically as $\beta\rightarrow\infty$,
the left hand side can be neglected yielding

$A_{\beta}\rightarrow\frac{dW}{dt}.$ (2.13)

After the transient time $t\gg 1/\beta$, $\langle A_{\beta}(t)\rangle\approx
0$ and $\langle
A_{\beta}(t)A_{\beta}(t^{\prime})\rangle\approx\beta\exp(\beta|t-t^{\prime}|)$
where $\langle\cdot\rangle$ denotes expactation. Thus, $\tau=1/\beta$ is the
correlation time. If $x_{\beta}$ is driven by $A_{\beta}$ as in

$dx_{\beta}=b(x_{\beta},t)dt+\epsilon A_{\beta}dt$ (2.14)

then as $\beta\rightarrow\infty$, $x_{\beta}$ converges to the solution of

$dx=b(x,t)dt+\epsilon dW$ (2.15)

where $x_{\beta}(0)=x(0)$. This can be made mathematically rigorous [2, 7]
Although more general colored noises are possible, I will restrict to those
described by eq. 2.12 in this note for simplicity.

Suppose a quantum process $x_{q}(t)$ is associated with a solution of
Schrödinger equation through Nelson’s formulation. An equivalent
characterization is the pair $(b(x,t),\rho(x,0))$ (one can solve for
$x_{q}(t)$ using eq. 2.4). Now replace the white noise in eq. 2.4 with colored
noise $A_{\beta}$ to define the new process

$dx(t)=b(x(t),t)dt+\epsilon A_{\beta}(t)dt$ (2.16)

where $\epsilon=\sqrt{\hbar/m}$, with the same initial density $\rho(x,0)$. As
$\beta\rightarrow\infty$, this should be a good approximation to $x_{q}(t)$.
However the stochastic acceleration of $x(t)$ will violate Nelson’s second
law:

$\frac{1}{2}(D_{+}D_{-}+D_{-}D_{+})x(t)\neq-\frac{1}{m}\frac{\partial
U(x)}{\partial x}|_{x(t)}$ (2.17)

as can be seen by direct calculation (see next section). The violation of
Nelson’s second law is of the same order of the quantum potential. As $x(t)$
and $x_{q}(t)$ are close to each other for large $\beta$ in an appropriate
norm, their stochastic derivatives differ. Of course, this is common as even
two deterministic functions with wildly differing derivatives can be close in
norm (e.g. take a reasonably smooth function and consider a function rapidly
fluctuating around it). This is why I call $x(t)$ as a colored noise smoothing
of $x_{q}(t)$.

If on the other hand, we consider the white noise limit of $A_{\beta}$ and
calculate small correlation time $\tau=1/\beta$ expansion, we get a process of
the form [8]

$dx_{\beta}(t)=b(x_{\beta}(t),t)dt+\epsilon(1+\tau g(x_{\beta}(t),t))dW(t)$
(2.18)

for some $g(x,t)$ related to $b(x,t)$. Now we see that for large $\beta$,
$x_{\beta}$ approximately satisfies Nelson’s second law and it is also close
to $x_{q}(t)$. The singular nature of the white noise limit is evident as
$D_{+}$ and $D_{-}$ are sensitive to the presence of Wiener terms as can be
seen from the definitions (eqs. 2.10 and 2.11).

###  2.3 Calculation of stochastic acceleration for a colored noise process

Consider the process of the form for given $b(x,t)$:

$\displaystyle dx(t)=b(x(t),t)dt+\epsilon A_{\beta}(t)dt$ (2.19)
$\displaystyle dA_{\beta}(t)=-\beta A_{\beta}(t)dt+\beta dW(t)$

where initially $x$ and $A_{\beta}$ are uncorrelated. We will make use of the
following formulas for forward and backward derivatives conditioned on fixed
$(x(t),A_{\beta}(t))$ of a function $G(x,A_{\beta},t)$ which can be found in
section 5 of Guerra’s review[4]:

$\displaystyle(D_{+}G)(x,A_{\beta},t)|_{(x(t),A_{\beta}(t))}$
$\displaystyle=\lim_{\Delta t\to 0^{+}}E[\frac{G(x(t+\Delta
t),A_{\beta}(t+\Delta t),t+\Delta t)-G(x(t),A_{\beta}(t),t)}{\Delta
t}|x(t),A_{\beta}(t)]$ (2.20) $\displaystyle=\frac{\partial G}{\partial
t}+(b+\epsilon A_{\beta})\frac{\partial G}{\partial x}-\beta
A_{\beta}\frac{\partial G}{\partial
A_{\beta}}+\frac{\beta^{2}}{2}\frac{\partial^{2}G}{\partial A_{\beta}^{2}}$

$\displaystyle(D_{-}G)(x,A_{\beta},t)|_{(x(t),A_{\beta}(t))}$
$\displaystyle=\lim_{\Delta t\to
0^{+}}E[\frac{G(x(t),A_{\beta}(t),t)-G(x(t-\Delta t),A_{\beta}(t-\Delta
t),t-\Delta t)}{\Delta t}|x(t),A_{\beta}(t)]$ (2.21)
$\displaystyle=\frac{\partial G}{\partial t}+(b+\epsilon
A_{\beta})\frac{\partial G}{\partial x}+(-\beta
A_{\beta}-\beta^{2}\frac{\partial}{\partial
A_{\beta}}\log\rho(x,A_{\beta},t))\frac{\partial G}{\partial
A_{\beta}}-\frac{\beta^{2}}{2}\frac{\partial^{2}G}{\partial A_{\beta}^{2}}$

We also need the following result on conditional expectations for a set of
random variables $(x,A,z)$:

$E[F(z)|x]=\int E[F(z)|A,x]p(A|x)dA$ (2.22)

for any function $F(z)$. We see that
$D_{+}x=D_{-}x|_{(x,A_{\beta})}=b+A_{\beta}$. Then

$D_{-}D_{+}x|_{(x,A_{\beta})}=D_{-}(b+A_{\beta})|_{(x,A_{\beta})}=\frac{\partial
b}{\partial t}+(b+\epsilon A\beta)\frac{\partial b}{\partial x}-\beta
A_{\beta}-\beta^{2}\frac{\partial}{\partial A_{\beta}}\log\rho(x,A_{\beta},t)$
(2.23)

and

$D_{+}D_{-}x|_{(x,A_{\beta})}=D_{+}(b+A_{\beta})|_{(x,A_{\beta})}=\frac{\partial
b}{\partial t}+(b+\epsilon A_{\beta})\frac{\partial b}{\partial x}-\beta
A_{\beta}$ (2.24)

Using eq. 2.22, we calculate the stochastic acceleration of $x(t)$ as

$\frac{1}{2}(D_{+}D_{-}+D_{-}D_{+})x(t)|_{x(t)}=\frac{\partial b}{\partial
t}+b\frac{\partial b}{\partial x}+(\epsilon\frac{\partial b}{\partial
x}-\beta)\int A_{\beta}\rho_{t}(A_{\beta}|x)dA_{\beta}$ (2.25)

For large $\beta$, $A_{\beta}$ is close to white noise so we can assume
$\rho_{t}(A_{\beta}|x)\approx\rho_{t}(A_{\beta})$ and neglect the last term
after the transient time as $\langle A_{\beta}\rangle\approx 0$. Thus we see
that Nelson’s second law is violated if $b(x,t)$ corresponds to a quantum
process.

###  2.4 A phase space derivation?

Now consider the phase space process

$\displaystyle dx_{\beta}(t)=v_{\beta}(t)dt+\epsilon A_{\beta}(t)dt$ (2.26)
$\displaystyle dv_{\beta}(t)=a(x_{\beta}(t))dt$ $\displaystyle
dA_{\beta}(t)=-\beta A_{\beta}(t)dt+\beta dW(t)$

where $v_{\beta}(t)$ is the velocity of the particle and $a(x_{\beta}(t))$ is
the acceleration on the particle. One can show through a similar calculation
as last section that Nelson’s second law is satisfied by $x_{\beta}(t)$ for
large but finite $\beta$. Also for large $\beta$, integrating out $v_{\beta}$
and $A_{\beta}$, $x_{\beta}(t)$ approximately satisfies

$dx_{\beta}(t)=(\int v\rho_{t}(v|x_{\beta})dv)dt+\epsilon A_{\beta}(t)dt$
(2.27)

where we identify $b(x,t)=\int v\rho_{t}(v|x)dv$ as the drift velocity. Hence,
$x_{\beta}(t)$ also approximately satisfies Nelson’s first law. Can we say
that as $\beta$ goes large then $x_{\beta}(t)$ becomes a quantum process
satisfying both Nelson’s first and second law exactly? More precisely, given
$x_{\beta}(t)$ satisfying eq. 2.26 for large $\beta$, does there exist a
quantum process $x_{q}(t)$ near $x_{\beta}(t)$? Conversely for a given quantum
process, does there exist $x(t)$ near it satisfying eq. 2.26? I don’t know the
answers to these questions. Here is the subtlety similar to that we
encountered in the colored noise smoothing of a quantum process. In the
singular limit when we replace $A_{\beta}(t)$ by white noise we have

$\displaystyle dx_{\infty}(t)=v_{\infty}(t)dt+\epsilon dW(t)$ (2.28)
$\displaystyle dv_{\infty}(t)=a(x_{\infty}(t))dt$

If we calculate the stochastic acceleration we see that due to the presence of
the Wiener term, extra terms are induced. Thus $x_{\infty}(t)$ violates the
second law. We can trace this back to the ordering of limits $\Delta
t\rightarrow 0$ in the definition of $D_{+}$ and $D_{-}$ and
$\beta\rightarrow\infty$. In general

$(D_{+}D_{-}+D_{-}D_{+})\lim_{\beta\rightarrow\infty}x_{\beta}(t)\neq\lim_{\beta\rightarrow\infty}(D_{+}D_{-}+D_{-}D_{+})x_{\beta}(t).$
(2.29)

What have we learned until now? We know that given a quantum process we can
construct a colored noise smoothing close to it albeit violating Nelson’s
second law. If we make a correlation time expansion around the singular white
noise limit, we get a process close to the quantum process and approximately
satisfying Nelson’s second law. The phase space process on the other hand
exactly satisfies Nelson’s second law for all finite and large $\beta$ and
approximates Nelson’s first law. However in the singular limit, it fails to
satisfy Nelson’s second law. Therefore it is not obvious whether the questions
raised above have affirmative answers. Hence I formulate two conjectures:

Conjecture 1. Let $(x(t),v(t),A_{\beta}(t))$ satisfy eq. 2.26. Assume that
$A_{\beta}$ is initially uncorrelated to $x$ and $v$. Then as $\beta$ becomes
large, there exists a quantum process $x_{q}(t)$, satisfying both of Nelson’s
laws (equivalent to a solution of the Schrödinger equation), which is close to
$x(t)$ projection of the phase space process in some appropriate norm.

Conjecture 2. Converse of Conjecture 1. Let $x_{q}(t)$ be a quantum process.
Then there exist a phase space process of the form given by eq. 2.26, which is
close to $x_{q}(t)$ in some appropriate norm, as $\beta$ becomes large.

I will argue in the following section that Conjecture 1 implies Conjecture 2.

###  2.5 Conjecture 1 implies Conjecture 2

Given a solution $\psi(x,t)$ to the Schrödinger equation, split it into
probability and phase functions as
$\psi(x,t)=\sqrt{\rho(x,t)}\exp(\frac{i}{\hbar}S(x,t))$. Then $\rho(x,t)$ and
$S(x,t)$ satisfy the Madelung equations:

$\frac{\partial\rho}{\partial t}=-\frac{\partial}{\partial
x}(\rho\frac{1}{m}\frac{\partial S}{\partial x})$ (2.30)

$\frac{\partial S}{\partial t}=-\frac{1}{2m}(\frac{\partial S}{\partial
x})^{2}-U(x)+\frac{\hbar^{2}}{2m}\frac{1}{\sqrt{\rho}}\frac{\partial^{2}}{\partial
x^{2}}\sqrt{\rho}$ (2.31)

We define $b(x,t)$ as

$b(x,t)=\frac{1}{m}\frac{\partial}{\partial
x}S(x,t)+\frac{\hbar}{2m}\frac{\partial}{\partial x}\log\rho(x,t).$ (2.32)

Then $\rho(x,t)$ satisfies

$\frac{\partial\rho(x,t)}{\partial t}=-\frac{\partial}{\partial
x}(b(x,t)\rho(x,t))+\frac{\hbar}{2m}\frac{\partial^{2}}{\partial
x^{2}}\rho(x,t)$ (2.33)

which is equivalent to Nelson’s first postulate:

$dx(t)=b(x(t),t)dt+\sqrt{\frac{\hbar}{m}}dW(t).$ (2.34)

Therefore given a solution to the Schrödinger equation, the associated Nelson
process is completely specified by $\rho(x,t)$ and $b(x,t)$. We would like to
construct an approximate solution to eq.2.26 to match with the Fokker-Planck
equation (eq.2.33). Since $\rho(x,t)$ is already fixed by the solution to the
Schrödinger equation, the only free function is $\hat{\rho}(v|x,t)$ as these
two determine the joint density
$\hat{\rho}(x,v,t)=\hat{\rho}(v|x,t)\rho(x,t)$. The mean of
$\hat{\rho}(v|x,t)$ is fixed by $b(x,t)$ (eq.2.27) which is determined from
the solution of the Schrödinger equation via eq.2.32. Furthermore, we have the
natural probability normalization constraint $\int\hat{\rho}(v|x,t)dv=1$.
Therefore, the problem becomes the following: Given a solution to the
Schrödinger equation represented by $\rho(x,t)$ and $b(x,t)$, construct an
approximate solution $\hat{\rho}(x,v,t)$ (construct $\hat{\rho}(v|x,t)$) to
eq.2.26 such that $\hat{\rho}(x,v,t)=\hat{\rho}(v|x,t)\rho(x,t)$, $b(x,t)=\int
v\hat{\rho}(v|x,t)dv$ and $\int\hat{\rho}(v|x,t)dv=1$. To construct a
solution, choose the initial conditional density $\hat{\rho}(v|x,t=0)$ such
that

$\int v\hat{\rho}(v|x,t=0)dv=b(x,t=0).$ (2.35)

Assuming conjecture 1 is true, the solutions to eq.2.26 is close to $x_{q}(t)$
satisfying Nelson’s both laws. Therefore, eq.2.26 approximately propagates the
initial $(\rho(x,t=0),b(x,t=0))$ to $(\hat{\rho}(x,t),\hat{b}(x,t))$ according
to the Schrödinger equation. Since $(\rho(x,t=0),b(x,t=0))$ completely
specifies the initial conditions for the Schrödinger equation (up to an
additive constant in the initial phase $S(x,t=0)$), the solutions are unique.
Therefore $(\hat{\rho}(x,t),\hat{b}(x,t))=(\rho(x,t),b(x,t))$. Hence given a
solution to the Schrödinger equation specified by $\rho(x,t)$ and $b(x,t)$, we
see that any initial density $\hat{\rho}(x,v,t=0)$ satisfying eq.2.35 and
$\int\hat{\rho}(x,v,t=0)dv=\rho(x,t=0)$ yields an approximate solution
$\hat{\rho}(x,v,t)$ to eq.2.26 such that $\int v\hat{\rho}(v|x,t)dv=b(x,t)$
and $\int\hat{\rho}(x,v,t)dv=\rho(x,t)$. Note that there are infinitely many
ways to choose the initial conditional density $\hat{\rho}(v|x,t=0)$
compatible with eq.2.35.

##  3 Multiple particles

###  3.1 Nelson’s mechanics for multiple particles

We have worked with a single particle until now. In this section, I give a
review of Nelson’s stochastic formulation of non-relativistic quantum
mechanics for $n$ particles with position variables
$\vec{x}=(x_{1},...,x_{n})$ and masses $m_{1},...,m_{n}$, interacting via the
potential $U(\vec{x})$. Consider the Schrödinger equation:

$i\hbar\frac{\partial\psi(\vec{x},t)}{\partial
t}=(-\sum_{j=1}^{n}\frac{\hbar^{2}}{2m_{j}}\frac{\partial^{2}}{\partial
x_{j}^{2}}+U(\vec{x}))\psi(\vec{x},t).$ (3.1)

Putting
$\psi(\vec{x},t)=\sqrt{\rho(\vec{x},t)}e^{\frac{i}{\hbar}S(\vec{x},t)}$, we
get the Madelung equations:

$\frac{\partial\rho(\vec{x},t)}{\partial
t}=-\sum_{j=1}^{n}\frac{\partial}{\partial
x_{j}}(\rho(\vec{x},t)\frac{1}{m_{j}}\frac{\partial S(\vec{x},t)}{\partial
x_{j}})$ (3.2)

$\frac{\partial S(\vec{x},t)}{\partial
t}=-\sum_{j=1}^{n}\frac{1}{2m_{j}}(\frac{\partial S(\vec{x},t)}{\partial
x_{j}})^{2}-U(\vec{x})+\sum_{j=1}^{n}\frac{\hbar^{2}}{2m_{j}}\frac{1}{\sqrt{\rho(\vec{x},t)}}\frac{\partial^{2}}{\partial
x_{j}^{2}}\sqrt{\rho(\vec{x},t)}$ (3.3)

where $\rho(\vec{x},t)$ is the probability of finding the particles at
$\vec{x}$ at time $t$ and $S(\vec{x},t)$ is the phase of the wave function. We
recognize first of the equations as the continuity equation with velocity
$(\frac{1}{m_{1}}\frac{\partial S}{\partial
x_{1}},...,\frac{1}{m_{n}}\frac{\partial S}{\partial x_{n}})$. The second of
the equations apart from the last term (quantum potential) on the right hand
side is the Hamilton-Jacobi equation. Thus if $\hbar=0$, we have the classical
ensemble of particles. We start by assuming that a particles obey the
following stochastic differential equations:

$dx_{i}(t)=b_{i}(\vec{x}(t),t)dt+\sqrt{\frac{\hbar}{m_{i}}}dW_{i}(t)$ (3.4)

where each $b_{i}(\vec{x},t)$ is a general function of positions and time, and
$(dW_{1},...,dW_{n})$ are independent Wiener processes. We call this as
Nelson’s first law or postulate. The diffusion equation associated to this
is[5, 6]

$\frac{\partial\rho(\vec{x},t)}{\partial
t}=-\sum_{j=1}^{n}\frac{\partial}{\partial
x_{j}}(b_{j}(\vec{x},t)\rho(\vec{x},t))+\sum_{j=1}^{n}\frac{\hbar}{2m_{j}}\frac{\partial^{2}}{\partial
x_{j}^{2}}\rho(\vec{x},t)$ (3.5)

where $\rho(\vec{x},t)$ is the probability of finding the particles at
$\vec{x}$ at time $t$. In order to match with the continuity equation, we
define

$\frac{\partial}{\partial
x_{i}}S(\vec{x},t)=m_{i}(b_{i}(\vec{x},t)-\frac{\hbar}{2m_{i}}\frac{\partial}{\partial
x_{i}}\log\rho(\vec{x},t))$ (3.6)

where we assumed that $\rho(\vec{x},t)$ is nowhere zero. For a discussion of
what happens at zeros, see Section 2. Note that in eq.3.6, we assume that
$(b_{1},...,b_{n})$ is a gradient. At first sight, this seems like an
independent assumption. However, it can be shown that the gradient assumption
is implied by the Newton-Nelson law using a variational argument (see Theorem
6 of [9] and [3]). Newton-Nelson law is introduced below. We want
$S(\vec{x},t)$ just defined in this way to satisfy the quantum Hamilton-Jacobi
equation. The quantum Hamilton-Jacobi equation can be shown to be equivalent
to the following equations:

$\frac{1}{2}(D_{+}D_{-}+D_{-}D_{+})x_{i}(t)=-\frac{1}{m_{i}}\frac{\partial
U(\vec{x})}{\partial x_{i}}|_{\vec{x}(t)}$ (3.7)

where $D_{+}$ and $D_{-}$ are forward and backward derivatives which will be
defined below, the right hand side is the classical acceleration of the
particle evaluated on the stochastic trajectory and the left hand side is the
time-symmetric stochastic acceleration. This is the stochastic analogue of
Newton’s second law. Thus we call this as Newton-Nelson law or Nelson’s second
law. The forward and backward derivatives are defined to be

$D_{+}x_{i}(t)=\lim_{\Delta t\to 0^{+}}E[\frac{x_{i}(t+\Delta
t)-x_{i}(t)}{\Delta t}|\vec{x}(t)],$ (3.8)

$D_{-}x_{i}(t)=\lim_{\Delta t\to 0^{+}}E[\frac{x_{i}(t)-x_{i}(t-\Delta
t)}{\Delta t}|\vec{x}(t)]$ (3.9)

where $E[f|\vec{x}(t)]$ denotes the expectation of $f$ conditioned on
$\vec{x}(t)$. For any function $F(\vec{x},t)$ we can write its forward and
backward derivatives explicitly as follows

$(D_{+}F)(\vec{x},t)=\frac{\partial}{\partial
t}F(\vec{x},t)+\sum_{j=1}^{n}b_{j}(\vec{x},t)\frac{\partial}{\partial
x_{j}}F(\vec{x},t)+\sum_{j=1}^{n}\frac{\hbar}{2m_{j}}\frac{\partial^{2}}{\partial
x_{j}^{2}}F(\vec{x},t),$ (3.10)

$(D_{-}F)(\vec{x},t)=\frac{\partial}{\partial
t}F(\vec{x},t)+\sum_{j=1}^{n}(b_{j}(\vec{x},t)-\frac{\hbar}{m_{j}}\frac{\partial}{\partial
x_{j}}\log\rho(\vec{x},t))\frac{\partial}{\partial
x_{j}}F(\vec{x},t)-\sum_{j=1}^{n}\frac{\hbar}{2m_{j}}\frac{\partial^{2}}{\partial
x_{j}^{2}}F(\vec{x},t).$ (3.11)

Using these formulas, it is straightforward to show that the Newton-Nelson law
is equivalent to the $\vec{x}$ derivative of the second Madelung equation
(eq.3.3). It has been shown that for each solution of the Schrödinger
equation, there is an associated stochastic process satisfying Nelson’s
postulates, and if Nelson’s postulates are satisfied, that one can construct a
wave function which satisfies the Schrödinger equation, with its absolute
square the probability density of the positions of particles [1, 2, 3].

###  3.2 A phase space derivation for multiple particles?

The multi-particle case is a straightforward generalization of the single
particle case. Consider the multi-particle phase space process

$\displaystyle dx_{i}(t)=v_{i}(t)dt+\epsilon A_{i}(t)dt$ (3.12) $\displaystyle
dv_{i}(t)=a(x_{1}(t),...,x_{n}(t))dt$ $\displaystyle
dA_{i}(t)=-\beta_{i}A_{i}(t)dt+\beta_{i}dW_{i}(t)$

where $x_{i}$, $v_{i}$,
$a(x_{i})=-(1/m_{i})\partial_{x_{i}}U(x_{1},...,x_{n},t)$ are the position,
velocity and the acceleration of particle $i$. Like the single particle case,
one can show that Newton-Nelson law is satisfied for large $\beta_{i}$. Also
integrating out the noises and velocities, we can approximate $x_{i}(t)$ by

$dx_{i}(t)=(\int v_{i}\rho_{t}(v_{i}|x_{1},...,x_{n})dv_{i})dt+\epsilon
A_{i}dt$ (3.13)

identifying $b_{i}(x_{1},....,x_{n},t)=\int
v_{i}\rho_{t}(v_{i}|x_{1},...,x_{n})dv_{i}$ as the drift velocity for particle
$i$. Here we assumed that close to the white noise limit with large
$\beta_{i}$, $A_{i_{1}}$ and $A_{i_{2}}$ become independent for $i_{1}\neq
i_{2}$. Subtleties similar to that in the single particle case remain here at
the white noise limit in calculating the stochastic acceleration. I formulate
conjectures which parallel the single particle case.

Conjecture 3. Let $\\{(x_{i}(t),v_{i}(t),A_{i}(t))\\}_{i}$ satisfy eq. 3.12.
Assume that $A_{i}$ is initially uncorrelated to $\vec{x}$ and $\vec{v}$. Then
as $\\{\beta_{i}\\}_{i}$ become large, there exists a quantum process
$\vec{x}_{q}(t)$, satisfying both of Nelson’s laws (equivalent to a solution
of the Schrödinger equation), which is close to $\vec{x}(t)$ projection of the
phase space process in some appropriate norm.

Conjecture 4. Converse of Conjecture 1. Let $\vec{x}(t)$ be a quantum process.
Then there exist a phase space process of the form given by eq. 3.12, which is
close to $\vec{x}_{q}(t)$ in some appropriate norm, as $\\{\beta_{i}\\}_{i}$
become large.

Next, I argue that Conjecture 3 implies Conjecture 4 with by a similar
argument given for the single particle case.

###  3.3 Conjecture 3 implies Conjecture 4

Given a solution $\psi(\vec{x},t)$ to the Schrödinger equation we split it
into probability and phase functions as
$\psi(\vec{x},t)=\sqrt{\rho(\vec{x},t)}\exp(\frac{i}{\hbar}S(\vec{x},t))$.
Then $\rho(\vec{x},t)$ and $S(\vec{x},t)$ satisfy the Madelung equations:

$\frac{\partial\rho}{\partial t}=-\sum_{i=1}^{n}\frac{\partial}{\partial
x_{i}}(\rho\frac{1}{m_{i}}\frac{\partial S}{\partial x_{i}})$ (3.14)

$\frac{\partial S}{\partial t}=-\sum_{i=1}^{n}\frac{1}{2m_{i}}(\frac{\partial
S}{\partial
x_{i}})^{2}-U(\vec{x})+\sum_{i=1}^{n}\frac{\hbar^{2}}{2m_{i}}\frac{1}{\sqrt{\rho}}\frac{\partial^{2}}{\partial
x_{i}^{2}}\sqrt{\rho}$ (3.15)

where the potential is
$U(x)=\sum_{i=1}^{n}\frac{m_{i}}{2}\omega_{i}^{2}x_{i}^{2}$ for $n$ non-
interacting particles. We define $b_{i}(x,t)$ as

$b_{i}(\vec{x},t)=\frac{1}{m_{i}}\frac{\partial}{\partial
x_{i}}S(\vec{x},t)+\frac{\hbar}{2m_{i}}\frac{\partial}{\partial
x_{i}}\log\rho(\vec{x},t).$ (3.16)

Then $\rho(x,t)$ satisfies

$\frac{\partial\rho(\vec{x},t)}{\partial
t}=-\sum_{i=1}^{n}\frac{\partial}{\partial
x_{i}}(b_{i}(\vec{x},t)\rho(\vec{x},t))+\sum_{i=1}^{n}\frac{\hbar}{2m_{i}}\frac{\partial^{2}}{\partial
x_{i}^{2}}\rho(\vec{x},t)$ (3.17)

which is equivalent to Nelson’s first postulate:

$dx_{i}(t)=b_{i}(\vec{x}(t),t)dt+\sqrt{\frac{\hbar}{m_{i}}}dW_{i}(t)$ (3.18)

for each $i$. Therefore, given a solution to the Schrödinger equation, the
associated Nelson process is completely specified by $\rho(\vec{x},t)$ and
$(b_{1}(\vec{x},t),...,b_{n}(\vec{x},t))$. We would like to construct an
approximate solution to eq.4.1 to match with the Fokker-Planck equation
(eq.3.17). Since $\rho(\vec{x},t)$ is already fixed by the solution to the
Schrödinger equation, the only free function is
$\hat{\rho}(\vec{v}|\vec{x},t)$, as these two determine the joint density
$\hat{\rho}(\vec{x},\vec{v},t)=\hat{\rho}(\vec{v}|\vec{x},t)\rho(\vec{x},t)$.
The mean of $\hat{\rho}(\vec{v}|\vec{x},t)$ is fixed by
$(b_{1}(\vec{x},t),...,b_{n}(\vec{x},t))$, which is determined from the
solution to the Schrödinger equation via eq.3.16. Furthermore, we have the
natural probability normalization constraint
$\int\hat{\rho}(\vec{v}|\vec{x},t)dv_{1}...dv_{n}=1$. Therefore the problem
becomes the following: Given a solution to the Schrödinger equation
represented by $\rho(\vec{x},t)$ and
$(b_{1}(\vec{x},t),...,b_{n}(\vec{x},t))$, construct an approximate solution
$\hat{\rho}(\vec{x},\vec{v},t)$ (construct $\hat{\rho}(\vec{v}|\vec{x},t)$) to
eq.4.1 such that
$\hat{\rho}(\vec{x},\vec{v},t)=\hat{\rho}(\vec{v}|\vec{x},t)\rho(\vec{x},t)$,
$b_{i}(\vec{x},t)=\int v_{i}\hat{\rho}(\vec{v}|\vec{x},t)dv_{1}...dv_{n}$ and
$\int\hat{\rho}(\vec{v}|\vec{x},t)dv_{1}...dv_{n}=1$. To construct a solution,
choose the initial conditional density $\hat{\rho}(\vec{v}|\vec{x},t=0)$ such
that

$\int v_{i}\hat{\rho}(\vec{v}|\vec{x},t=0)dv_{1}...dv_{n}=b_{i}(\vec{x},t=0)$
(3.19)

for each $i$. Assuming Conjecture 3 is true, $\vec{x}$ projections of the
solutions to eq.4.1 are close to a quantum process $\vec{x}_{q}(t)$ satisfying
both of Nelson’s laws. Therefore eq.4.1 approximately propagates the initial
$(\rho(\vec{x},t=0),b_{1}(\vec{x},t=0),...,b_{n}(\vec{x},t=0))$ to
$(\rho(\vec{x},t),b_{1}(\vec{x},t),...,b_{n}(\vec{x},t))$ according to the
Schrödinger equation. Since
$(\rho(\vec{x},t=0),b_{1}(\vec{x},t=0),...,b_{n}(\vec{x},t=0))$ completely
specifies the initial conditions for the Schrödinger equation (up to an
additive constant in the initial phase $S(\vec{x},t=0)$), the solutions are
unique. Therefore
$(\hat{\rho}(\vec{x},t),\hat{b}_{1}(\vec{x},t),...,\hat{b}_{n}(\vec{x},t))=(\rho(\vec{x},t),b_{1}(\vec{x},t),...,b_{1}(\vec{x},t))$.
Hence given a solution to the Schrödinger equation specified by
$\rho(\vec{x},t)$ and $(b_{1}(\vec{x},t),...,b_{n}(\vec{x},t))$, we see that
any initial density $\hat{\rho}(\vec{x},\vec{v},t=0)$ satisfying eq.3.19 and
$\int\hat{\rho}(\vec{x},\vec{v},t=0)dv_{1}...dv_{n}=\rho(\vec{x},t=0)$ yields
an approximate solution $\hat{\rho}(\vec{x},\vec{v},t)$ to eq.4.1 such that
$\int v_{i}\hat{\rho}(\vec{v}|\vec{x},t)dv_{1}...dv_{n}=b_{i}(\vec{x},t)$ for
each $i$ and
$\int\hat{\rho}(\vec{x},\vec{v},t)dv_{1}...dv_{n}=\rho(\vec{x},t)$. Note that
there are infinitely many choices of the initial conditional density
$\hat{\rho}(\vec{v}|\vec{x},t=0)$ compatible with eq.3.19.

##  4 Locality and local causality

###  4.1 Nonlocality of Nelson’s theory vs locality of phase space model

The multi-particle Nelson’s theory is manifestly non-local as is well known
[3]. It is easy to see that the drift velocity $b_{i}$ of each particle
depends on the positions of all other particles in general. Even if the
particles are no longer interacting, a potential acting on one particle
affects the drift velocities of other particles. In this sense, Nelson’s
formulation is non-local. As argued by Nelson in [3] this could be due to the
Markovian nature of noise and in a non-Markovian theory non-locality may not
be required as non-locality could be traded off with temporal correlations.
The phase space model presented here is non-Markovian due to the colored
nature of the noises introduced. Is it also local? Assume that Conjectures 3
and 4 hold. The following simple argument is in favor of locality. Let the
particles interact for some finite time until $t=0$ and we turn off the
interactions (keeping single particle potentials) and separate the particles.
Close to the white noise limit each particle obeys the equation

$\displaystyle dx_{i}(t)=v_{i}(t)dt+\epsilon A_{i}(t)dt$ (4.1) $\displaystyle
dv_{i}(t)=a_{i}(x_{i}(t))dt$

As $A_{i}$’s will be uncorrelated for large $\beta_{i}$’s, every particle is
subject to independent forces and independent potentials. The model is local
in this sense. However the initial correlations can be preserved and this
would give rise to quantum correlations. When the velocities are integrated
out, we would obtain Nelson’s theory which is non-local. The non-locality
arises from integrating out degrees of freedom. We know from Bell type
inequalities that certain theories cannot simulate quantum theory. Assuming
that the measurement settings are independent from the variables of the
theory, since the phase space is local, if it is simulate quantum theory, it
cannot be locally causal [10]. In the next sections, I elaborate on this:
local stochastic theories need not to be locally causal, therefore Bell-type
inequalities provide no obstacle against the existence of a local stochastic
underlying theory for quantum mechanics which is itself not locally causal.

###  4.2 Multi-time measurements in stochastic mechanics

In order to make contact with Bell-type inequalities, we need to review how to
make sense of multi-time measurements in Nelson’s theory. It is well known
that for position measurements at a single time, the predictions of quantum
mechanics and that of stochastic mechanics agree [1]. It is quite easy to see
this is the case. The quantum mechanical expectation value of functions $f$
and $g$ of the positions $x_{1}$ and $x_{2}$ at time $t$ is given by

$\int\rho(x_{1},x_{2},t)f(x_{1})g(x_{2})dx_{1}dx_{2},$ (4.2)

which is the same as that given by stochastic mechanics. Since the Schrödinger
equation and stochastic equations agree on the evolution of
$\rho(x_{1},x_{2},t)$, the quantum expectation value is equal to the classical
expectation value, therefore the theories are completely equivalent when
position measurements are considered at a single time. However, it is also
well known that the predictions of stochastic mechanics differ from that of
quantum mechanics for measurements of commuting position variables at
different times. The quantum mechanical expectation value of functions $f$ and
$g$ of positions $x_{1}$ at time $t_{1}$ and $x_{2}$ at time $t_{2}$ is given
by

$\int\rho(x_{1},x_{2},t_{1})\rho(\tilde{x}_{1},\tilde{x}_{2},t_{2}|x_{1},x_{2},t_{1})f(x_{1})g(\tilde{x_{2}})dx_{1}dx_{2}d\tilde{x}_{1}d\tilde{x}_{2}$
(4.3)

where $\rho(\tilde{x}_{1},\tilde{x}_{2},t_{2}|x_{1},x_{2},t_{2})$ is the
quantum mechanical probability obtained from solving the Schrödinger equation,
for the particles reaching to $(\tilde{x_{1}},\tilde{x}_{2})$, at time $t_{2}$
given that their positions were $(x_{1},x_{2})$ at time $t_{1}$. In general,
this multi-time expectation value contradicts the expectation value $\langle
x_{1}(t_{1})x_{2}(t_{2})\rangle$ obtained in stochastic mechanics. All these
generalize to multiple particles and measurements at more than $2$ times, or
multi-time measurements.

For multi-time measurements, in order to make the quantum mechanical
expectation values compatible with the expectation values calculated from
stochastic mechanics, Blanchard et al [11] proposed that when the stochastic
mechanical expectation values are computed, the probability density must be
updated upon measurement. This process is the stochastic mechanical
counterpart of the wave-function collapse. To be concrete, consider the two
particle example. Assume that $t_{2}>t_{1}$. When $x_{1}$ is measured at time
$t_{1}$, the following reduction takes place in the probability density:
$\rho(x_{1},x_{2},t_{1})$ is replaced by
$\rho_{m}(x_{1},x_{2},t_{1}|\bar{x}_{1})=\rho(x_{2}|x_{1},t_{1})\delta(x_{1}-\bar{x}_{1})$,
where $\bar{x}_{1}$ is the measured value of $x_{1}$. The stochastic equations
are propagated with the new initial density
$\rho_{m}(x_{1},x_{2},t_{1}|\bar{x}_{1})$ to
$\rho_{m}(x_{1},x_{2},t_{2}|\bar{x}_{1})$. When the expectation values are
computed, one first compute averages fixing $\bar{x}_{1}$, then averages over
the density $\rho(\bar{x}_{1},t_{1})=\rho(x_{1}=\bar{x}_{1},t_{1})$. This
gives the quantum mechanical expectation value (eq. 4.3) [11]. However, if one
does not take into account the probability density reduction and propagates
the initial density $\rho(x_{1},x_{2},t_{2})$, one would get correlations in
stochastic mechanics which would contradict the quantum expectation values. It
must be stressed that this property is very peculiar to stochastic mechanics.
In a usual diffusion process which is governed by a partial differential
equation linear in probability, collapsing the initial probability,
propagating the solution to a later time and averaging over the initial
probability gives the same results for the statistics when the initial density
is directly propagated. However, in stochastic mechanics, the drift velocities
$b_{i}$ depend on the probability density, hence the diffusions are non-
linear. When the probability is updated upon measurement and propagated, the
drift velocities would be different from the the ones obtained by propagating
the initial density directly. It is this non-linearity of stochastic mechanics
seems to enable the probability collapse to make the quantum and stochastic
mechanical expectation values equal for multi-time position measurements. For
details, see [11].

Is the above probability collapse scheme non-local? Assume Conjecture 3 and 4
hold. Nelson’s stochastic mechanics is manifestly non-local: anything done on
the first particle will be felt by the second particle instantly because of
the drift $b_{2}$ of the second particle depends on the position $x_{1}$ of
the first particle in a general entangled state. The non-locality of Nelson’s
mechanics emerges from integrating out the velocity variables of the phase
model, which is local. If one takes the point of view that the probabilities
only represent the knowledge that one has about the underlying particle
trajectories, there is nothing non-local in terms of how the particles evolve.
It is the probability distribution that collapses, which is not part of
ontology. The real quantities, which are the phase space stochastic
trajectories of the particles, are not affected by each other, as is clear
from that they are dynamically uncoupled and the random forces driving them
are independent.

How to make contact with Bell type theorems? Consider the two particle case.
Suppose that Albert has the two particles with phase coordinates
$(x_{1},v_{1})$ and $(x_{2},v_{2})$ and he prepares them in the state
$\rho_{0}(x_{1},v_{1},x_{2},v_{2})$, which corresponds to an entangled quantum
states, by some local procedure. Then he sends the first particle to John and
the second particle to Edward. Let John and Edward synchronize their clocks.
John makes a measurement of position $x_{1}$ with outcome $\bar{x}_{1}$ at
time $t_{1}$ and Edward makes a measurement of position $x_{2}$ with outcome
$\bar{x}_{2}$ at a later time $t_{2}$. After this procedure is repeated many
times, John and Edward report back their measurement outcomes to Albert for
him to calculate empirical expectation values of functions of $\bar{x}_{1}$
and $\bar{x}_{2}$. Since $\rho(x_{1},x_{2},t)$ evolves according to the
Schrödinger equation, and since the quantum Heisenberg position operators
$X_{1}(t_{1})$ and $X_{2}(t_{2})$ commute, the quantum mechanical expectation
value $\langle f(X_{1}(t_{1}))g(X_{2}(t_{2}))\rangle$ is given by eq.4.3. This
must coincide with the empirical expectation values that Albert has computed
since the particles obey the Schrödinger equation. How would Albert compute
expectation values in the stochastic model? He has two major choices. Either
for each experiment, he assumes that John measured $x_{1}$ at $t_{1}$ and it
yielded $\bar{x}_{1}$ and he updates the probability density to make a refined
estimate of Edward’s measurement of $x_{2}$ at $t_{2}$ conditioned on
$\bar{x}_{1}$, then he averages over the probability density
$\rho(\bar{x}_{1})$, or he calculate the expectations directly without
collapsing the probability distribution. We discussed earlier in this section
that these would in general give different results. If Albert chooses to
update the probability density of two particles upon measurement, then he
would obtain the quantum mechanical expectation values. Therefore, we assume
that Albert calculate the expectation values in this way to make sure that
multi-time measurements in quantum theory and stochastic mechanics coincide
without violating locality.

###  4.3 Local causality

With the above measurement prescription, we obtain a correspondence between
stochastic mechanics and quantum theory with multi-time measurements. Assuming
Conjectures 3 and 4 are true and the measurement settings are independent from
the particle variables (non-conspiring), this provides a local stochastic
theory for quantum theory. Bell-type theorems [10] imply that local, locally
causal and non-conspiring deterministic or stochastic hidden variable theories
satisfy inequalities that are violated by quantum mechanics. As quantum
mechanics is not locally causal [10], neither the phase space model can be if
the conjectures 3 and 4 are true.

Although local deterministic theories are locally causal, these two versions
of locality are separate for stochastic theories. There is no reason that a
local stochastic theory be locally causal automatically. See chapter 23 of
[3]. To elaborate on this, lets remind ourselves what locality and local
causality means in the sense of Bell [10]. Locality means that influences
cannot propagate with a speed greater than that of light. This is the usual
relativistic notion of locality. Local causality is subtler. Let $R1$ and $R2$
be spacelike separated regions. Let $A$($B$) be stochastic variables (beables)
in $R1$($R2$). Denote the beables of the past of $R1$ ($R2$) minus the
intersection of pasts of $R1$ and $R2$ as $N$($M$). Denote the beables in the
intersections of the pasts of $R1$ and $R2$ as $\Lambda$. Local causality
states that $p(A|\Lambda,N,B)=p(A|\Lambda,N)$: the beables at $R2$ does not
affect the outcomes at space-like seperated $R1$ given the past beables
$\Lambda$ and $N$ of $R1$. This is of course true in deterministic local
theory as the past of $R1$ determines what happens at $R1$ completely. In a
stochastic theory where the dynamics is deterministic and initial conditions
are unknown, local causality still holds. However, in general local causality
is independent from locality for stochastic theories [3]. A sufficiently
chaotic theory can cause loss of information from the past of $R1$ to $R1$,
still retaining useful information in $R2$ as correlations. This chaotic
dynamics can emerge from deterministic dynamics of many degrees of freedom or
it could be due to external noise. For a general stochastic theory,
information in $R2$ could be non-redundant given the past of $R1$ without any
causal connection between $R1$ and $R2$. Hence if conjectures 3 and 4 are to
be true, the local phase space model cannot be locally causal.

##  5 Fields

###  5.1 Potential physical sources of noise

How physical is to assume that the noise appears as velocity rather than a
force in the phase space process? A possible answer to this comes from
consideration of canonical momentum rather than kinematic momentum. A similar
problem appears when one quantize a charged non-relativistic particle in an
electromagnetic field. We know that canonical commutation relations should be
imposed on canonical rather than kinematic variables. Consider the Lagrangian
of a unit charged particle with unit mass with position and velocity $(x,v)$
in an electromagnetic field $(\phi,A)$:

$L=\frac{1}{2}v^{2}-\phi(x,t)+v\cdot A$ (5.1)

The canonical momentum is

$p=\frac{\partial L}{\partial v_{i}}=v_{i}+A_{i}$ (5.2)

and the equation of motion is

$\dot{p_{i}}=\frac{\partial L}{\partial x_{i}}=-\frac{\partial\phi}{\partial
x_{i}}=a_{i}$ (5.3)

With the canonical variables $x,p$ we have

$\displaystyle dx_{i}=(p_{i}-A_{i})dt$ (5.4) $\displaystyle dp_{i}=a_{i}dt$

Restricting to one dimension we have

$\displaystyle dx=pdt-Adt$ (5.5) $\displaystyle dp=a(x)dt$

Now we see that if the electromagnetic potential $A$ is the colored noise
$A_{\beta}$, we have induced noise as velocity. On the other hand if we have
used kinematic momentum $v$, we would not have the noise term as velocity but
acceleration. Having the noise term as a velocity seems to be important to
make direct contact with Nelson’s first law.

A similar argument can be given for a non-relativistic particle in a weak
gravitational field where again the canonical and kinematic momenta differ and
the source of noise would be gravitational. In the next sections, I provide a
quantization of the free scalar field in a stochastic gravitational field
assuming conjectures 3 and 4.

###  5.2 Stochastic quantization of the free scalar field

The phase space model for multiple particles enables a generalization to
fields. The simplest case is the free scalar field though interacting fields
and fields with spin could be treated in similar ways with some technical
complications. First, I review stochastic quantization of the free scalar
field following [4]. Consider the Klein-Gordon field $\phi$ satisfying

$\partial_{t}^{2}\phi-\Delta\phi+m^{2}\phi=0$ (5.6)

where $\Delta$ is the three dimensional Laplace operator. Put the system in a
box $B$ with appropriate boundary conditions with a real complete orthonormal
basis of eigenfunctions $u_{i}$ of $\Delta$ satisfying

$\int_{B}u_{i}(x)u_{j}(x)d^{3}x=\delta_{ij}$ (5.7)

where $\sum_{i}u_{i}(x)u_{i}(x^{\prime})=\delta_{B}(x-x^{\prime})$ (delta
function in $B$: $\delta_{B}(x-x^{\prime})\rightarrow\delta(x-x^{\prime})$ in
the infinite volume limit) and $\Delta u_{i}(x)=-k_{i}^{2}u_{i}(x)$. Expand
$\phi$ in the form

$\phi(x,t)=\sum_{i}q_{i}(t)u_{i}(x)$ (5.8)

or

$q_{i}(t)=\int_{B}\phi(x,t)u_{i}(x)d^{3}x.$ (5.9)

Then each $q_{i}(t)$ is a harmonic oscillator with frequency
$\omega_{i}=\sqrt{m^{2}+k_{i}^{2}}$:

$\frac{d^{2}q_{i}}{dt^{2}}+\omega_{i}^{2}q_{i}=0.$ (5.10)

Assume the Hamiltonian of the field is
$\int\frac{1}{2}((\partial_{t}\phi)^{2}+\partial_{i}\phi\partial^{i}\phi+m^{2}\phi^{2})d^{3}x$.
To stochastically quantize the field, we promote each of the harmonic
oscillators to quantum oscillators unit mass (not to be confused with the
field mass $m$):

$dq_{i}(t)=b_{i}(q_{1}(t),...,q_{n}(t),t)dt+\epsilon dW_{i}(t)$ (5.11)

and

$\frac{1}{2}(D_{+}D_{-}+D_{-}D_{+})q_{i}(t)=-\omega_{i}^{2}q_{i}(t).$ (5.12)

where $dW_{i}(t)$’s are independent Wiener processes and
$\epsilon=\sqrt{\hbar}$. Does the noise $\\{dW_{i}(t)\\}_{i}$ act locally on
$\phi$? To see this, calculate the stochastic derivative of $\phi$ using
eq.5.8:

$d\phi(x,t)=\sum_{i}b_{i}(q_{1}(t),...,q_{n}(t),t)u_{i}(x)dt+\epsilon\sum_{i}u_{i}(x)dW_{i}(t)$
(5.13)

for each $x\in B$. The noise term is again Gaussian since it is a linear sum
of Gaussian processes with mean and covariance

$\epsilon\langle\sum_{i}u_{i}(x)dW_{i}(t)\rangle=0$ (5.14)

$\epsilon^{2}\langle\sum_{i}u_{i}(x)dW_{i}(t)\sum_{j}u_{j}(x^{\prime})dW_{j}(t)\rangle=\epsilon^{2}\sum_{i}u_{i}(x)u_{i}(x^{\prime})dt=\epsilon^{2}\delta(x-x^{\prime})dt$
(5.15)

in the infinite volume limit where we used $\langle dW_{i}(t)\rangle=0$ and
$\langle dW_{i}(t)dW_{j}(t)\rangle=\delta_{ij}dt$. Hence, we can write

$d\phi(x,t)=\sum_{i}b_{i}(q_{1}(t),...,q_{n}(t),t)u_{i}(x)dt+\epsilon
dW(x,t).$ (5.16)

with $\langle dW(x,t)\rangle=0$ and $\langle
dW(x,t)dW(x^{\prime},t)\rangle=\delta(x-x^{\prime})dt$. In this sense, the
noise acts locally on $\phi(x,t)$. Note that each
$b_{i}(q_{1}(t),...,q_{n}(t),t)$ can be written in terms of $\phi(x,t)$ using
eq.5.9. Nelson’s second law has a simple expression in terms of for the field:

$\frac{1}{2}(D_{+}D_{-}+D_{-}D_{+})\phi(x,t)=(\Delta-m^{2})\phi(x,t)$ (5.17)

as can be seen using eq.5.8 and eq.5.12. This can be also obtained by
replacing the acceleration term $\partial_{t}^{2}\phi$ by its stochastic
counterpart $1/2(D_{+}D_{-}+D_{-}D_{+})\phi$ in eq.5.6. Does this stochastic
quantization depend on the time slicing? Yes, it is obviously dependent on
coordinates. For a different time slicing, one gets a different stochastic
process. All these stochastic processes together should preserve the
relativistic covariance of the resulting quantum theory [4].

Now, I give a phase space model for the stochastic field. Instead of eq.5.11,
we have

$\displaystyle dq_{i}=v_{i}dt+\epsilon A_{i}dt$ (5.18) $\displaystyle
dv_{i}=-\omega_{i}^{2}q_{i}dt$ $\displaystyle dA_{i}=-\beta_{i}A_{i}dt+\beta
dW_{i}$

as we had for the multi-particle case. Assuming Conjectures 3 and 4 are true,
for large $\beta$, this provides a phase space model for the stochastic field.
Is this model local? To see this, write the above equations in terms of
$\phi(x,t)$ using eq.5.8 and defining $V(x,t)=\sum_{i}v_{i}(t)u_{i}(x)$:

$\displaystyle d\phi(x,t)=V(x,t)dt+\epsilon\sum_{i}u_{i}(x)A_{i}(t)dt$ (5.19)
$\displaystyle dV(x,t)=(\Delta-m^{2})\phi(x,t)dt$ $\displaystyle
dA_{i}=-\beta_{i}A_{i}dt+\beta dW_{i}$

for each $x\in B$. In the large $\beta_{i}$ limit, assuming $A_{i}$’s should
become independent Wiener processes. Thus, as above,
$\sum_{i}u_{i}(x)A_{i}(t)$ converges to a Wiener process with zero mean and
covariance $\delta(x-x^{\prime})dt$ in the infinite volume limit. The
dynamical equation $dV(x,t)=(\Delta-m^{2})\phi(x,t)dt$ is already local.
Locality of $d\phi(x,t)=V(x,t)dt+\epsilon\sum_{i}u_{i}(x)A_{i}(t)$ is obtained
in the infinite volume and large $\beta_{i}$ limit.

###  5.3 Gravitational origins of noise

Can the source of noise be gravitational? I outline a model below. Consider
the scalar field $\phi$ in a weak stochastic background gravitational field
$g_{\mu\nu}$ with the action

$S=\frac{1}{2}\int
d^{4}x\sqrt{-g}(g_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi-m^{2}\phi^{2}-2\xi
R\phi).$ (5.20)

where $R$ is the Ricci scalar, $g=\text{det}g$ and
$\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$. Putting
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, expanding $S$ to first order in
$h_{\mu\nu}$ and integrating the non-minimal coupling term $\xi R\phi$ by
parts we have

$S=\frac{1}{2}\int
d^{4}x[\partial^{\mu}\phi\partial_{\mu}\phi-m^{2}\phi^{2}-h^{\mu\nu}(\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\eta_{\mu\nu}\partial^{\sigma}\phi\partial_{\sigma}\phi)+2\xi(\partial_{\nu}h^{\mu\nu}-\partial^{\mu}h^{\sigma}_{\sigma})\partial_{\mu}\phi]$
(5.21)

The canonical momentum of the field is

$\Pi=\frac{\delta S}{\delta(\partial^{0}\phi)}=\partial_{0}\phi-
h_{0\nu}\partial^{\nu}\phi+\frac{1}{2}h^{\sigma}_{\sigma}\partial_{0}\phi+\xi(\partial^{\nu}h_{0\nu}-\partial_{0}h).$
(5.22)

In the quantum regime assume that the minimal coupling part can be neglected
as compared to the non-minimal coupling part. Of course for the experimentally
tested classical regime, the minimal coupling part should dominate the non-
minimal part. We have in the quantum regime

$\partial_{0}\phi=\Pi-\xi(\partial^{\nu}h_{0\nu}-\partial_{0}h).$ (5.23)

The acceleration equation can be obtained from the Euler-Lagrange equation
yielding

$\partial_{0}\Pi=\Delta\phi-m^{2}\phi+F^{\mu\nu}(\phi,\partial_{\sigma}\phi)h_{\mu\nu}$
(5.24)

where $F^{\mu\nu}$ is some linear differential operator depending on $\phi$
and $\partial_{\sigma}\phi$. In the Newtonian gauge
$h_{\mu\nu}=\text{diag}(\theta,\theta,\theta,\theta)$ assuming some isotropy
condition on the stress-energy tensor generating it. Hence, we have the system
of equations

$\displaystyle\partial_{0}\phi=\Pi+\xi\partial_{0}\theta$ (5.25)
$\displaystyle\partial_{0}\Pi=\Delta\phi-m^{2}\phi+G(\phi,\partial_{\sigma}\phi)\theta.$

In order to stochastically quantize the field, we need to make the
identification (assuming conjectures 3 and 4): $\xi\partial_{0}\theta=\epsilon
A_{\beta}$ where $A_{\beta}$ is a space-time colored noise, which is
characterized by some set of parameters $\beta$, converging to space-time
white noise (time derivative of the zero mean Wiener process with covariance
$\delta(x-x^{\prime})dt$ as above in the infinite volume limit) as $\beta$
becomes large. Assuming that the statistics of $\theta$ becomes independent of
$\phi$ when $\beta$ is large and $\langle\theta\rangle=0$, eq.5.25 yields
Nelson’s second law (eq.5.17). This finalizes the sketch of a phase space
quantization of the scalar field in a stochastic background gravitational
field. Note that, this cannot be achieved with minimal coupling in the
framework considered here since the noise terms induced from minimal coupling
would depend on the field itself.

Can this background gravitational field be sourced by a matter distribution?
In the limit of large $\beta$, in the infinite volume limit, $\theta$ is
characterized by a space-time Gaussian field with zero mean and covariance:

$\langle
d\theta(x,t)d\theta(x^{\prime},t)\rangle=\frac{\epsilon^{2}}{\xi^{2}}\delta(x-x^{\prime})dt$
(5.26)

In a Newtonian universe, assume small fluctuations around a constant mean
(dark) matter density. A zero mean homogeneous isotropic Gaussian matter
distribution can be described by the two-point function of the fluctuating
part of the matter density $\delta\rho(x)$:

$c(x-x^{\prime},t)=\langle\delta\rho(x,t)\delta\rho(x^{\prime},t)\rangle$
(5.27)

whose Fourier transform $P(k)$ is a power law:

$P(k,t)=\int c(x,t)e^{-ik\cdot x}d^{3}x=A(t)|k|^{n}.$ (5.28)

The spectrum of the gravitational potential $\theta$ can be calculated from
the Poisson equation:

$\Delta\phi=4\pi G\delta\rho$ (5.29)

where $G$ is the Newton’s constant. Taking the Fourier transform of both sides
we can show that the spectrum $P_{\theta}(k)$ of $\theta$ is given by

$P_{\theta}(k,t)=\frac{(4\pi G)^{2}}{|k|^{4}}P(k,t).$ (5.30)

The Fourier transform of the covariance of $\theta$ is:

$P_{\theta}(k,t)=\frac{\epsilon^{2}}{\xi^{2}}t$ (5.31)

from which we obtain

$P(k,t)=\frac{|k|^{4}}{(4\pi G)^{2}}\frac{\epsilon^{2}}{\xi^{2}}t.$ (5.32)

Analogous expressions can be written for the case of expanding universe. Of
course, this is a tall order: fluctuations should remain up until where
quantum field theory is tested in particle accelerators. Also, Planck’s
constant is emergent and can depend on time in principle. Note that the matter
distribution should temporally fluctuate as white noise. This is consistent
assuming that the fluctuations arise from the gravitational interactions of
all particles (fields) in the universe as Brownian motion of single particles
emerge from the microscopic dynamics in a gas or plasma [12].

## References

  * [1] E. Nelson.  Derivation of the Schrödinger equation from Newtonian mechanics.  Phys. Rev., 150:1079–1085, Oct 1966.
  * [2] E. Nelson.  Dynamical Theories of Brownian motion.  Princeton Univesity Press, 1967.
  * [3] E. Nelson.  Quantum fluctuations.  Princeton Univesity Press, 1985.
  * [4] F. Guerra.  Structural aspects of stochastic mechanics and stochastic field theory.  Physics Reports, 77:263–312, 1981.
  * [5] A. Friedman.  Stochastic differential equations and applications.  Academic Press, 1975.
  * [6] L. Arnold.  Stochastic differential equations: theory and applications.  John Wiley and Sons, 1974.
  * [7] G. A. Pavliotis and A. M. Stuart.  Multiscale methods: averaging and homogenization.  Springer Science, 2008.
  * [8] Peter Häunggi and Peter Jung.  Colored Noise in Dynamical Systems, pages 239–326.  John Wiley and Sons, Ltd, 1994.
  * [9] K. Yasue.  Stochastic calculus of variations.  Journal of Functional Analysis, 41(3):327 – 340, 1981.
  * [10] J. S. Bell and Alain Aspect.  Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy.  Cambridge University Press, 2 edition, 2004.
  * [11] P. Blanchard, S. Golin, and M. Serva.  Repeated measurements in stochastic mechanics.  Phys. Rev. D, 34:3732–3738, Dec 1986.
  * [12] F. Calogero.  Cosmic origin of quantization.  Physics Letters A, 228(6):335 – 346, 1997.


# Positive pressure matters in acoustic droplet vaporization

Samuele Fiorini  Anunay Prasanna  Gazendra Shakya  Marco Cattaneo  Outi
Supponen  Institute of Fluid Dynamics, ETH Zürich, Zürich, Switzerland

###### Abstract

Acoustically vaporizable droplets are phase-change agents that can improve the
effectiveness of ultrasound-based therapies. In this study, we demonstrate
that the compression part of an acoustic wave can generate tension that
initiates the vaporization. This counter-intuitive process is explained by the
occurrence of Gouy phase shift due to the focusing of the acoustic wave inside
the droplet. Our analysis unifies the existing theories for acoustic droplet
vaporization under a single framework and is supported by experiments and
simulations. We use our theory to identify governing parameters that allow to
vaporize droplets using predominantly compression waves, which are safer in
medical use.

_Introduction._ Micron- and sub-micron-sized superheated liquid
perfluorocarbon droplets are a relatively new class of ultrasound-responsive
agents that are typically metastable in blood circulation and can be converted
into microbubbles upon acoustic excitation [1], with the possibility to target
specific areas of the human body. This process, commonly known as acoustic
droplet vaporization (ADV), shows promise in applications such as contrast-
enhanced ultrasound imaging [2], gas embolotherapy [3], targeted drug delivery
[4], and other ablation techniques [5, 6]. Compared to microbubble contrast
agents, droplet emulsions present greater stability against dissolution in the
blood flow [7] and can be nanometric in size to leverage the Enhanced
Permeability and Retention (EPR) effect to improve drug accumulation in tumor
tissue [8]. Vaporization inception is typically characterized by an acoustic
threshold, which depends non-trivially on parameters such as droplet radius,
frequency, and temperature [9].

Significant progress has been made in elucidating the physics of ADV [10, 11,
12, 13, 14], as well as in characterizing and reducing the acoustic
vaporization threshold [1, 15, 16], which currently greatly exceeds the
pressure required to excite microbubble contrast agents [17] and can question
the safety of droplet-based _in vivo_ treatments. Shpak _et al._ [12]
suggested that the focusing of superharmonics of the driving wave, created due
to nonlinear propagation in the medium surrounding the droplets, can generate
regions of localized tension amplification in the liquid core that facilitate
nucleation. They experimentally showed that this effect is particularly
pronounced for large micrometric droplets (radius $>4$
$\mathrm{\SIUnitSymbolMicro m}$), and further supported their results by means
of theoretical modeling. Lajoinie _et al._ [18] described another possible
mechanism for droplet vaporization based on acoustic resonance. By solving a
linearized equation for the droplet radial dynamics, they theoretically
predicted the resonance frequency and the pressure amplification factor. The
calculated maximum amplification factor for a resonant pressure field is
practically constant with respect to droplet radius, making acoustic resonance
a good candidate to exploit for vaporizing sub-micron droplets.

While most of the prior studies on ADV focus on the peak negative pressure
(PNP) as the main contributor to vaporization initiation, in this Letter we
demonstrate that the compressive phase of the driving acoustic wave can play a
significant role in triggering phase-change by means of a physical mechanism
that has been previously overlooked. Specifically, it can dictate the location
of the maximum tension within the droplet bulk due to the occurrence of _Gouy
Phase Shift_ [19], which leads to a sign reversal of the focusing incoming
wave as it crosses the focal point. This new insight advances our
understanding of the physics behind ADV and the presented analysis unifies the
superharmonic focusing and resonance theories under the same theoretical
framework.

_Theory._ We model the axisymmetric acoustic wave propagation within a
spherical droplet of radius $R$, density $\rho_{2}$, and speed of sound
$c_{2}$, located in an infinite medium of density $\rho_{1}$ and speed of
sound $c_{1}$ as described elsewhere [12, 20, 21]. The transmitted pressure
within the droplet is obtained by summing the solutions for every harmonic
component, $n$, of the incoming nonlinear ultrasound wave with fundamental
angular frequency $\omega$ and initial phase $\phi$, owing to the linearity of
the wave equation. The pressure field in the droplet core can be described as
a function of the radial coordinate $r$, the polar angle $\theta$, and the
time $t$ as:

$\leavevmode\resizebox{216.81pt}{}{{\hbox{p_{t}(r, \theta, t) = \real\left[
\sum_{n = 0}^{\infty} \sum_{m = 0}^{\infty} a_ne^{in\omega t +
\phi_n}\tilde{\alpha}_{m,n}j_m(nk_2r)P_m(\cos{\theta})\right]}}},$ (1)

where

$\leavevmode\resizebox{216.81pt}{}{{\hbox{\tilde{\alpha}_{m,n} = (-i)^m(2m +
1)\frac{j_{m}(nk_{1}R)h_{m}^{\prime(2)}(nk_{1}R) -
h_{m}^{(2)}(nk_{1}R)j_{m}^{\prime}(nk_{1}R)}{j_{m}(nk_{2}R)h_{m}^{\prime(2)}(nk_{1}R)
-
\frac{\rho_{1}c_{1}}{\rho_{2}c_{2}}h_{m}^{(2)}(nk_{1}R)j_{m}^{\prime}(nk_{2}R)}}}},$
(2)

is the amplification factor with $k_{1}=\omega/c_{1}$ the fundamental
wavenumber in the surrounding medium, $k_{2}=\omega/c_{2}$ the fundamental
wavenumber inside the droplet, $j_{m}$ and $y_{m}$ the spherical Bessel
functions of the first and second kind of order $m$, $h_{m}=j_{m}-iy_{m}$ the
spherical Hankel function of the second kind of order $m$ and the prime
represents differentiation with respect to $k_{q}r$, where $q=1,2$. The
complete derivation is provided in the Supplementary Material [22].

Figure 1:  (a) Schematic of the experimental setup. (b)  _Top_ : Snapshots of
the vaporization dynamics of a single 9-$\mathrm{\SIUnitSymbolMicro m}$-radius
droplet sonicated with a 5-$\mathrm{MHz}$ HIFU. _Below_ : Experimentally
observed nucleation positions (blue dots) overlapped with the numerically
simulated instantaneous pressure field on the droplet center plane when the
global minimum pressure is reached. (c) Similar to (b), with a
38-$\mathrm{\SIUnitSymbolMicro m}$-radius droplet and a 3.5-$\mathrm{MHz}$
HIFU. In both cases, nucleation inception happens at the time
$t=0\;$\mathrm{ns}$$.

_Numerical simulations._ Two-dimensional axisymmetric numerical simulations
are performed using the k-Wave toolbox in MATLAB [23]. The experimentally
measured pressure wave at the transducer focal point is used to calculate the
temporal profile of a velocity source in the $z$-direction and to create a
traveling plane wave in the simulation domain. The droplet is modeled as a
circular discontinuity in density and sound speed, placed in an infinite
homogeneous medium (water). The simulated pressure fields on the droplet
centerline show excellent agreement with the theory upon reaching steady-
state, thereby validating the theoretical model (see Supplementary Material
for details [22]).

_Experimental setup._ Monodisperse perfluoropentane (C5F12, PFP, ABCR Swiss
AG) droplets with an average radius of 9 $\mathrm{\SIUnitSymbolMicro m}$ and
38 $\mathrm{\SIUnitSymbolMicro m}$ are prepared using polydimethylsiloxane
(Elastosil RT601 A/B, Wacker Chemie AG) microfluidic chips fabricated in-house
following the protocols described elsewhere [24, 25]. A 5% v/v solution of
fluorosurfactant (Capstone™ FS-30, Apollo Scientific) is employed as the
continuous phase to ensure the long-term stability of the droplets.
Polydisperse droplet populations are prepared by emulsification of 1% v/v
fluorosurfactant solution and perfluoropentane using an amalgamator (MHC
Technology, FR).

The experimental setup employed to characterize ADV is illustrated in Fig.
1(a). Vaporization dynamics are visualized using a custom-built vertical
microscope [26] with a 3-$\mathrm{mm}$ focal length objective (Olympus
LumPlanFL N 60x water dipping) and the possibility to mount either a
600-$\mathrm{mm}$ focal length (TL600-A, Thorlabs) or a 200-$\mathrm{mm}$
focal length (TTL200-A, Thorlabs) tube lens, providing an overall
magnification of 200$\times$ and 66.66$\times$, respectively. The microscope
is coupled to a high-speed camera (Shimadzu HPV-X2) set to record at 10
million frames per second using backlight illumination provided by two xenon
flash lamps used sequentially (MVS-7010, PerkinElmer). The droplets are placed
within a 1% w/v agarose test chamber, submerged in a water tank maintained at
an ambient temperature of 37–40 $\mathrm{\SIUnitSymbolCelsius}$ by an
immersion heater. The droplets are subjected to acoustic excitation from a
5-$\mathrm{MHz}$ (Precision Acoustics, UK) and 3.5-$\mathrm{MHz}$ (Olympus,
JPN) single-element high-intensity focused ultrasound (HIFU) transducer,
driven by an arbitrary wave generator (LW420B, Teledyne LeCroy) and amplified
by a 53-$\mathrm{dB}$ linear power amplifier (1020L, E&I). A linear array
probe (LA332, Esaote, IT) controlled by an ultrasound advanced open platform
(ULA-OP, MSD Lab, Università degli Studi di Firenze, IT [27]) is additionally
tested to investigate the influence of wave distortion on the acoustic
threshold. The pressure field at the optical and acoustic focus is
experimentally measured using a 75-$\mathrm{\SIUnitSymbolMicro m}$ needle
hydrophone (NH0075, Precision Acoustics, UK).

_Results and discussion._ Experimental frames depicting the initial stage of
ADV are presented in Fig. 1(b) for a 9-$\mathrm{\SIUnitSymbolMicro m}$ and
Fig. 1(c) for a 38-$\mathrm{\SIUnitSymbolMicro m}$-radius droplet upon
excitation with a 5-$\mathrm{MHz}$ and 3.5-$\mathrm{MHz}$ HIFU, respectively
(the full recordings are available in the Supplementary Material [22]). For
both recordings, $t=0$ $\mathrm{ns}$ represents the time when the first
nucleated vapor bubble becomes visible. The numerically computed instantaneous
pressure field corresponding to the global pressure minimum is shown below the
images and compared with the experimentally observed nucleation locations. As
expected, the nucleation spots are positioned close to the region of global
minimum pressure, which in these specific conditions (acoustic wave, droplet
radius, temperature) is highly localized either within the proximal side of
the droplet or close to its center.

The free-field experimental pressure profile generated by the 5-$\mathrm{MHz}$
HIFU, used as an input for the theoretical model, is depicted in Fig. 2(a)
together with the theoretically evaluated pressure field along the centerline
of the 9-$\mathrm{\SIUnitSymbolMicro m}$-radius droplet. The experimental
pressure wave presents a marked asymmetry between the positive and negative
peaks due to the wave’s distortion upon nonlinear propagation in water and the
resulting presence of superharmonics. To investigate the effect of the
distortion on tension generation in the droplet core, the wave is split into
its compression and rarefaction parts, and the harmonic content of the
resulting monopolar waves, shown in Fig. 2(b), is then used to compute two
separate pressure fields. Similar to examining distinct harmonic components of
a wave, the linearity of the wave equation implies that the complete pressure
field within the droplet core can be recovered by summing these two partial
contributions.

Figs. 2(c)-(d) display two instantaneous pressure profiles corresponding to
the monopolar wave solutions at the instant of time in which the combination
of the two has a global pressure minimum. The rarefaction-induced pressure
field in Fig. 2(c) does not show a particularly localized focal region, but
rather a broad amplification of the wave traveling through the droplet core. A
maximum amplification of 1.1 times the incoming wave PNP is achieved at the
location corresponding to the global pressure minimum. On the other hand,
surprisingly, the compression-induced solution in Fig. 2(d) shows a highly
localized negative pressure peak with a maximum amplification factor of 1.05
with respect to the incoming wave PNP. The location of this localized negative
pressure peak on the distal side of the droplet matches well with the minimum
pressure achieved in the combined solution, suggesting that the compression
phase dictates the location of the latter in addition to contributing to its
absolute value.

Figure 2:  (a) Theoretically calculated pressure field along the symmetry axis
of a 9-$\mathrm{\SIUnitSymbolMicro m}$ radius droplet, compared with a single
cycle of the free-field, experimentally measured 5-$\mathrm{MHz}$ HIFU
pressure signal. (b) Harmonic content of the compression and rarefaction phase
of the experimentally measured pressure wave, normalized by the coefficient of
the fundamental harmonic of the complete wave. (c) Transmission of the
rarefaction and (d) compression part of the experimentally recorded wave
computed on the droplet centerline compared with the respective undisturbed
reference. The shaded area represents the droplet position. All snapshots
depict the time instant at which the minimum pressure of the full solution is
reached.

Since the acoustic impedance $Z=\rho c$ of PFP is lower than water, tension
generation cannot be attributed to the reflection of the monopolar compression
wave at the internal boundary of the droplet. Instead, it can be seen from
animations (see Supplementary Material [22]) that the monopolar compression
wave, once transmitted into the droplet core, converges and, at its focus,
develops a negative tail that is then reflected at the rear boundary and
refocused at the second focal position in the proximal side of the droplet.
Here, the negative pressure generated by the compression peak combines with
the amplified negative pressure obtained from the transmission of the
rarefaction phase, thus generating a global minimum throughout the whole
excitation period. We believe that a so-called _Gouy phase shift_ happens
while the compression wave crosses both focal points. Louis Georges Gouy first
reported this effect in 1890 [19] for electromagnetic waves, but this
principle can be applied to any wave subjected to transverse spatial
confinement [28] and has recently been experimentally verified for focusing
acoustic waves [29]. For a three-dimensional focusing beam, the Gouy phase
shift consists of an additional variation in phase of $-\pi$ with respect to a
plane wave traveling in the same medium, gradually acquired as the wave
crosses the focal point [30] and corresponding to a sign inversion.

Due to the high longitudinal spatial confinement within the droplet, classical
modeling of the Gouy phase shift, which relies on the fact that the wave
travels in an infinite medium [31, 32, 30, 28], cannot be directly applied. To
assess the phase shift of monochromatic plane waves with different
frequencies, we performed a numerical simulation in which a plane wave with a
Gaussian temporal profile $f(t)=e^{(t-\mu)/2\sigma}$
($\mu=3.3\;$\mathrm{ns}$$, $\sigma=0.55\;$\mathrm{ns}$$) is transmitted and
focused in a 9-$\mathrm{\SIUnitSymbolMicro m}$-radius droplet. Three snapshots
are represented in Fig. 3(a) at the times $t_{f}\pm 5.6\;$\mathrm{ns}$$, with
$t_{f}$ being the time at which the wave reaches the focal point. Most of the
harmonics of interest contained in the focusing wave can be analyzed by
performing a Fourier transform of the signal inside the droplet core at
different time instants (i.e., different positions of the propagating wave in
the droplet core). For each selected instant and each of the constitutive
Fourier harmonics, the phase acquired by a plane wave traveling in the same
medium, $\Phi_{n}^{ref}=2\pi f_{n}\Delta t=2\pi\lambda_{n}\Delta z$, can be
deducted to estimate the Gouy phase, following a procedure similar to the one
found in Zhu _et al._ [32]. Here, $f_{n}$ and $\lambda_{n}$ represent the
frequency and wavelength of the $n$-th harmonic, and $\Delta t=\Delta z/c_{2}$
represents the time needed for the wave to travel a distance $\Delta z$ in the
droplet. The Gouy phase is set to equal zero at the instant the wave reaches
its focus, $\Phi_{G}(z^{\prime}=0)=0$. We calculated a phase shift of around
$\pi$ across the focal point, shown in Fig. 3(b), for the harmonic components
of the incoming wave with frequency $f\simeq 20~{}$\mathrm{MHz}$$ and above.
Calculating this for lower frequencies is not possible due to the Fourier
transform resolution limitation $\Delta f=c_{2}/2R\simeq 20~{}$\mathrm{MHz}$$.
Moreover, monochromatic plane waves with frequencies lower than 20
$\mathrm{MHz}$ (i.e. wavelengths larger than the droplet diameter), do not
show focusing as predicted by geometrical acoustics but present a maximum at
the droplet center, as the focusing wave interacts with its reflection at the
droplet wall within the span of its wavelength.

Figure 3:  (a) Selected instants of the pressure field computed at the
centerline of a droplet due to interaction with a plane wave having a purely
compressive Gaussian temporal profile. The time at which the weave reaches the
focal point is $t_{f}$, while $\Delta t=5.6\ $\mathrm{ns}$$. (b) Calculated
phase shift of different frequency components of a Gaussian wave focusing
inside a 9-$\mathrm{\SIUnitSymbolMicro m}$-radius droplet. In both cases,
$z^{\prime}=z-z_{f}$ represents the distance from the droplet focal point
$z_{f}$.

To better describe this behavior, acoustic mode decomposition can be employed.
From Eq. (1) it is clear that every harmonic component of the driving wave
excites an infinite number of acoustic modes of the droplet, represented by
Bessel functions of the first kind multiplied by Legendre polynomials. Every
acoustic mode is multiplied by a gain factor $\tilde{\alpha}_{m}$ which, in
dimensionless variables, depends on the ratio between density
$\rho_{1}/\rho_{2}$ and speed of sound $c_{1}/c_{2}$ of the droplet and
surrounding medium, the order of the considered mode $m$, and the
dimensionless droplet radius $\widetilde{R}=fR/c_{2}$. The dependency on the
harmonic index $n$ is incorporated in the generic frequency, $f$. Assuming
fixed material properties, the problem can be compactly described by
$\widetilde{R}$, which includes the combined effect of the frequency of the
driving wave and the droplet radius. Curves of the amplification factors
$\tilde{\alpha}_{m}$ for the acoustic modes of order $m$ are presented in Fig.
4(a), and are calculated using the physical properties for water and PFP.

Every mode presents multiple resonance frequencies in which the amplification
has a local maximum, as depicted in Fig. 4(a) for the mode $m=0$. For a
monochromatic wave, the full set of amplification factors required to
calculate the theoretical solution can be obtained by intersecting a vertical
line $\widetilde{R}=\bar{f}\overline{R}/c_{2}$ for fixed values of $R$ and
$f$, with the curves of the different $\tilde{\alpha}_{m}$. Modes with $m\geq
1$ share the property of being equal to zero at the droplet center, where
instead the mode $m=0$ presents a global maximum due to the shape of the
spherical Bessel functions. Therefore, if the zeroth mode is the most excited,
focusing of the wave in the classical sense, i.e., at the focal point
predicted by geometrical acoustics, is not clearly visible, and the droplet
will be in an “acoustic resonance” regime, similar to what is described by
Lajoinie _et al._ [18]. However, if higher modes are excited, two main focal
points will be visible, one on the distal and one on the proximal side of the
droplet due to successive reflection at the droplet wall. The droplet will be
in a “phase-shift” regime, with wave sign inversion occurring as it crosses
the focal points (distal and proximal). This can be interpreted as the
“superharmonic focusing” effect reported by Shpak _et al._ [12]. The
transition between the two regimes is gradual, but a conventional limit can be
fixed at the $\widetilde{R}$ value where the gain for the mode $m=1$ overcomes
the one for $m=0$, corresponding to $\widetilde{R}\simeq 0.4$, which is
consistent with $f\geq c_{2}/2R$ estimated for the effect of droplet
confinement on wave focusing. The limit is represented in Fig. 4(a) as a
vertical red line.

The theory is consistent with the experimental results presented beforehand.
The harmonic content of the 5-$\mathrm{MHz}$ wave presented in Fig. 2(b) shows
a negligible rarefaction component above 20 $\mathrm{MHz}$, which corresponds
to a dimensionless radius of $\overline{R}_{\mathrm{rar}}\backsim 0.44$ for a
9-$\mathrm{\SIUnitSymbolMicro m}$-radius droplet. As a result, the energy
content of the rarefaction phase will mostly excite the mode $m=0$. The
compression phase instead presents energy in the fifth harmonic (25
$\mathrm{MHz}$), corresponding to a value
$\overline{R}_{\mathrm{comp}}\backsim 0.55$ very close to the $m$ = 1
resonance peak, thus explaining the localization of the nucleation spots.
Similarly, the 3.5-$\mathrm{MHz}$ wave employed for the sonication of
38-$\mathrm{\SIUnitSymbolMicro m}$-radius droplets contains significant energy
in the first harmonic (both for the compression and rarefaction phase),
corresponding to $\overline{R}_{\mathrm{comp}}\backsim 0.33$. Despite the
calculated position of the pressure minimum being slightly away from the
droplet center, most likely due to the effect of higher harmonics in the
driving wave, the nucleation event is dominated by the resonance regime, as
demonstrated by the nucleation map in Fig. 1(c).

A direct implication of the wave’s positive pressure contribution to enhance
tension generation is a significant reduction of the vaporization threshold
$\text{PNP}_{\mathrm{th}}$, as shown in Fig. 4(b). This parameter is defined
by extracting the PNP of the driving wave for which single droplet
vaporization is first observed optically, obtained by gradually increasing the
energy supplied to the transducer. The apparent distinction between the
5-$\mathrm{MHz}$ HIFU and the linear array probe in the vaporization threshold
can be attributed to the different harmonic content of the two excitation
waves, with the HIFU wave in Fig. 4(c) being much less distorted than the one
driven by the linear array probe in Fig. 4(d) (see also Supplementary Material
[22]). Wave distortion enables significant tension generation with relatively
low PNP, facilitated by the phase-shift of the compression part of the driving
wave.

Once the critical value of the dimensionless radius has been set, the
function, $\widetilde{R}=0.4$, can be plotted on the frequency-radius space,
thus delimiting the “resonance” and the “phase-shift” regime on the $(f,R)$
space. The resulting graph is presented in Fig. 4(e), where the gray-shaded
area represents the necessary conditions for phase shift occurrence within the
droplet core. The harmonic content of the compression and rarefaction phases
is reported in correspondence with the radius and frequency tested in our
experiments and in selected past studies [12, 18]. This graph provides an
intuitive way to assess the possibility of “resonance” and “phase-shift” by
looking at the location of the harmonic content in the $(f,R)$ space. In
general, the presence of harmonics in the “phase-shift” regime indicates the
opportunity to achieve elevated tension, either due to the focusing of the
rarefaction part of the wave or the phase shift of the compression one. A more
accurate prediction would however require an assessment of the occurrence of
resonances by inspecting the $\widetilde{R}$ values for every harmonic of the
driving wave.

Figure 4:  (a) Amplification factor for the acoustic modes of order $m$ of a
PFP droplet surrounded by water. The red line represents the point at which
the amplification of the mode $m=1$ is higher than the amplification for the
zeroth order mode. (b) Vaporization PNP threshold of PFP droplets
experimentally observed for two different ultrasound transducers as a function
of droplet radius. The pressure measurements have an accuracy of $\pm 10\%$,
while the error on the radius measurements is estimated to be $\pm 0.32$
$\mathrm{\SIUnitSymbolMicro m}$. (c)-(d) Waveforms of the 5-$\mathrm{MHz}$
HIFU and the linear array probe driven at 5-$\mathrm{MHz}$, respectively. (e)
Harmonic content of the rarefaction and compression phase of the waves
employed in this study and some selected works from literature plotted on the
$(f,R)$ space. The gray area represents the $(f,R)$ value-pairs in the “phase-
shift” regime. The green area represents the clinically relevant operating
window, considering the maximum size allowed for intravenous agents and the
minimum frequency typically employed for ultrasound diagnostics [33].

_Conclusions._ The theory described here helps demonstrate the significance of
the compression phase of a distorted ultrasound wave in promoting ADV. The
main outcome lies in the finding that for a given radius, harmonics with a
frequency $f>0.4c_{2}/R$ present a pronounced phase shift while crossing the
focal point of the PFP droplet resulting in sign reversal of the waveform. The
distortion of a wave due to nonlinear propagation in the medium causes the
high-frequency harmonic content to concentrate in the compression phase of the
ultrasound, promoting the occurrence of Gouy phase shift at the focus and the
generation of a localized tension region. The proposed scaling using the
dimensionless radius allows extending the range of droplet sizes while
maintaining a similar pressure field with the practically relevant solutions
(i.e., green area in Fig. 4(e)), potentially solving the inherent
technological limitation of optical observation of sub-micron droplet
dynamics.

Our results suggest that describing the acoustic vaporization threshold of a
droplet solely by the applied PNP of the incoming wave is insufficient. The
full extent of the wave spectrum is rarely taken into account despite its
importance in determining the maximum tension generated in the droplet core,
and this could be a reason behind the discrepancies between the vaporization
thresholds reported by prior ADV studies [15, 17].

The occurrence of the phase shift offers the opportunity to optimize the
acoustic driving for micro- and nanodroplets by designing the shape of the
driving wave. Distorted waves with a weak rarefaction phase and a strong peak
positive pressure can generate the required tension in the droplet core
without affecting the surrounding medium, improving the safety of ADV while
still achieving the desired therapeutic effects.

_Acknowledgments._ This work was funded by the Swiss National Science
Foundation (SNSF project number 200021_200567). We thank Prof. Inge Herrmann
for lending us the $3.5$-$\mathrm{MHz}$ HIFU transducer.

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