Hamiltonian Formulation of the Pilot-Wave Theory

Dan N. Vollick

Irving K. Barber School of Arts and Sciences

University of British Columbia Okanagan

3333 University Way

Kelowna, B.C.

Canada

V1V 1V7

Abstract

In the pilot-wave theory of quantum mechanics particles have definite
positions and velocities and the system evolves deterministically. The
velocity of a particle is determined by the wave function of the system (the
guidance equation) and the wave function evolves according to Schrodinger’s
equation.

In this paper I first construct a Hamiltonian that gives Schrodinger’s
equation and the guidance equation for the particle. I then find the
Hamiltonian for a relativistic particle in Dirac’s theory and for a quantum
scalar field.

##  1 Introduction

In the standard approach to quantum mechanics the wave function provides a
complete description of any system and is used to calculate probability
distributions for observables associated with the system. In the pilot-wave
theory pioneered by de Broglie [1, 2] in the 1920’s, particles have definite
positions and velocities and are “guided” by the wave function, which
satisfies Schrodinger’s equation. A similar approach was developed by Bohm [3,
4] in the early 1950’s (see [5] and [6] for extensive reviews).

For a single particle in the non-relativistic theory the velocity of the
particle is given by

$\frac{d\vec{X}}{dt}=\vec{J}(\vec{X},t)\;,$ (1)

where $\vec{X}$ is the particle’s position and

$\vec{J}=\frac{\hbar}{2im|\psi|^{2}}\left[\psi^{*}\vec{\nabla}\psi-\psi\vec{\nabla}\psi^{*}\right]\;.$
(2)

Particle trajectories are, therefore, integral curves to the vector field
$\vec{J}$.

In this paper I first consider the pilot-wave theory for a non-relativistic
particle in a potential. I construct a Hamiltonian that gives Schrodinger’s
equation and the guidance equation for the particle. This involves imposing a
constraint and the use of Dirac’s method of dealing with constrained dynamical
systems [7, 8]. I then find the Hamiltonian for a relativistic particle in
Dirac’s theory and for a quantum scalar field.

A Hamiltonian formulation of the pilot-wave theory has also been developed by
Holland [9]. His approach is significantly different from the approach taken
in this paper and will be discussed at the end of section 2.

##  2 Non-Relativistic Pilot-Wave Theory

In this section I construct a Hamiltonian that gives Schrodinger’s equation
and the guidance equation for a single particle and for a collection of
particles.

The Lagrangian

$L=\frac{1}{2}i\hbar\left[\psi^{*}\frac{\partial\psi}{\partial
t}-\psi\frac{\partial\psi^{*}}{\partial
t}\right]-\frac{\hbar^{2}}{2m}\vec{\nabla}\psi\cdot\vec{\nabla}\psi^{*}-V\psi^{*}\psi$
(3)

gives Schrodinger’s equation

$i\hbar\frac{\partial\psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V\psi$ (4)

under a variation with respect to $\psi^{*}$ and its complex conjugate under a
variation with respect to $\psi$.

The canonical momenta $\Pi_{\psi}=\partial L/\partial\dot{\psi}$ and
$\Pi_{\psi^{*}}=\partial L/\partial\dot{\psi}^{*}$ are given by

$\Pi_{\psi}=\frac{1}{2}i\hbar\psi^{*}\hskip 28.90755ptand\hskip
28.90755pt\Pi_{\psi^{*}}=-\frac{1}{2}i\hbar\psi\;.$ (5)

We therefore have the primary constraints

$\phi_{1}=\Pi_{\psi}-\frac{1}{2}i\hbar\psi^{*}\approx 0\hskip
28.90755ptand\hskip
28.90755pt\phi_{2}=\Pi_{\psi^{*}}+\frac{1}{2}i\hbar\psi\approx 0\;,$ (6)

where $\approx$ denotes a weak equality, which can only be imposed after the
Poisson brackets have been evaluated. These constraints satisfy

$\\{\phi_{1}(\vec{x}),\phi_{1}(\vec{y})\\}=\\{\phi_{2}(\vec{x}),\phi_{2}(\vec{y})\\}=0$
(7)

and

$\\{\phi_{1}(\vec{x}),\phi_{2}(\vec{y})\\}=-i\hbar\delta^{3}(\vec{x}-\vec{y})\;.$
(8)

These constraints are, therefore, second class constraints.

The canonical Hamiltonian density
$h_{C}=\Pi_{\psi}\dot{\psi}+\Pi_{\psi^{*}}\dot{\psi}^{*}-L$ is given by

$h_{C}=\frac{\hbar^{2}}{2m}\vec{\nabla}\psi\cdot\vec{\nabla}\psi^{*}+V\psi^{*}\psi+\phi_{1}\dot{\psi}+\phi_{2}\dot{\psi}^{*}$
(9)

and the total Hamiltonian density is given by

$h_{T}=\frac{\hbar^{2}}{2m}\vec{\nabla}\psi\cdot\vec{\nabla}\psi^{*}+V\psi^{*}\psi+u_{1}\phi_{1}+u_{2}\phi_{2}\;,$
(10)

where $u_{1}$ and $u_{2}$ are undetermined parameters.

For consistency we require that

$\dot{\phi_{1}}=\\{\phi_{1},H_{T}\\}\approx 0\hskip 28.90755ptand\hskip
28.90755pt\dot{\phi}_{2}=\\{\phi_{2},H_{T}\\}\approx 0\;,$ (11)

where $H_{T}=\int h_{T}d^{3}x$ is the total Hamiltonian. These two equations
give

$u_{1}=\frac{i}{\hbar}\left[\frac{\hbar^{2}}{2m}\nabla^{2}\psi-V\psi\right]$
(12)

and

$u_{2}=-\frac{i}{\hbar}\left[\frac{\hbar^{2}}{2m}\nabla^{2}\psi^{*}-V\psi^{*}\right]\;.$
(13)

Substituting these expressions for $u_{1}$ and $u_{2}$ into $H_{T}$ and
integrating by parts gives

$H_{T}=\frac{i}{\hbar}\int\left[\Pi_{\psi}\left(\frac{\hbar^{2}}{2m}\nabla^{2}\psi-V\psi\right)-\Pi_{\psi^{*}}\left(\frac{\hbar^{2}}{2m}\nabla^{2}\psi^{*}-V\psi^{*}\right)\right]d^{3}x\;.$
(14)

The equation of motion for $\psi$ is

$\dot{\psi}=\\{\psi,H_{T}\\}=\frac{i}{\hbar}\left(\frac{\hbar^{2}}{2m}\nabla^{2}\psi-V\psi\right)\;,$
(15)

which is Schrodinger’s equation. The equation of motion for $\Pi_{\psi}$ is

$\dot{\Pi}_{\psi}=\\{\Pi_{\psi},H_{T}\\}=\frac{i}{\hbar}\left(\frac{\hbar^{2}}{2m}\nabla^{2}\Pi_{\psi}-V\Pi_{\psi}\right)\;.$
(16)

From (6) we see that this is Schrodinger’s equation for $\psi^{*}$. Similar
results hold for $\dot{\psi}^{*}$ and for $\dot{\Pi}_{\psi^{*}}$. The
constraints $\phi_{1}$ and $\phi_{2}$ can be promoted to strong equations if
the Poisson bracket is replaced by the Dirac bracket.

A similar Hamiltonian density
$h=-\frac{i\hbar}{2m}\vec{\nabla}\Pi\cdot\vec{\nabla}\psi-\frac{i}{\hbar}V\Pi\psi$
appears in Schiff [10], but he does not follow Dirac’s procedure to obtain
$h$. Gergely [11] follows Dirac’s approach and finds the Hamiltonian density
$h=\frac{\hbar^{2}}{2m}\vec{\nabla}\psi^{*}\cdot\vec{\nabla}\psi+V\psi^{*}\psi+\dot{\psi}\phi_{1}+\dot{\psi}^{*}\phi_{2}$.
Schrodinger’s equation and its complex conjugate then follow from the
conditions $\dot{\phi}_{1}\simeq 0$ and $\dot{\phi}_{2}\simeq 0$ (see [12] for
a discussion on the Hamiltonian formulation of the DKP equation, which is also
a first order equation, using Dirac’s procedure).

In the pilot-wave theory the wave function is written as

$\psi=Re^{iS/\hbar}\;,$ (17)

where $R$ and $S$ are real functions. The velocity of the particle is taken to
be [2, 3]

$\frac{d\vec{X}}{dt}=\frac{1}{m}\vec{\nabla}S(\vec{X},t)\;,$ (18)

where $\vec{X}(t)$ is the particle’s position. The velocity can also be
written as

$\frac{d\vec{X}}{dt}=\frac{\vec{j}}{\psi^{*}\psi}=\frac{\hbar}{2im|\psi|^{2}}\left[\psi^{*}\vec{\nabla}\psi-\psi\vec{\nabla}\psi^{*}\right]\;,$
(19)

where $\vec{j}$ is the probability current density.

This equation of motion follows from the Hamiltonian

$H_{p}=\frac{\vec{p}}{m}\cdot\vec{\nabla}S(\vec{X},t)\;.$ (20)

To see this consider the equations of motion

$\dot{X}^{k}=\\{X^{k},H_{p}\\}=\frac{1}{m}\left[\frac{\partial S}{\partial
x^{k}}\right]_{\vec{x}=\vec{X}}$ (21)

and

$\dot{p}_{k}=\\{p_{k},H_{p}\\}=-\frac{p_{l}}{m}\left[\frac{\partial^{2}S}{\partial
x^{k}\partial x^{l}}\right]_{\vec{x}=\vec{X}}\;.$ (22)

Equation (21) is the correct equation of motion for the particle. Equation
(22) gives the equation of motion for $p_{k}$, but $p_{k}$ is not related to
the particle velocity in this theory. We can therefore solve for the
particle’s trajectory using (21) alone and ignore (22).

Now consider the Hamiltonian for the field and particle

$H=H_{T}+H_{P}\;.$ (23)

The equations of motion $\dot{\psi}=\\{\psi,H\\}$, $\dot{X}^{k}=\\{X^{k},H\\}$
and $\dot{p}_{k}=\\{p_{k},H\\}$ are the same as above. However the equation of
motion for $\Pi_{\psi}$ is altered by the addition of $H_{p}$ to $H_{T}$:

$\dot{\Pi}_{\psi}=\\{\Pi_{\psi},H\\}=\\{\Pi_{\psi},H_{T}\\}+\frac{p_{k}}{m}\\{\Pi_{\psi},\partial_{k}S\\}\;.$
(24)

To evaluate $\\{\Pi_{\psi},\nabla^{k}S\\}$ it is convenient to write
$\partial_{k}S$ as

$\partial_{k}S(\vec{X},t)=\frac{\hbar}{2i}\int\left[\frac{\partial_{k}\psi(\vec{x},t)}{\psi(\vec{x},t)}-\frac{\partial_{k}\psi^{*}(\vec{x},t)}{\psi^{*}(\vec{x},t)}\right]\delta^{3}(\vec{x}-\vec{X})d^{3}x\;.$
(25)

The last term in (24) given by

$\frac{p_{k}}{m}\\{\Pi_{\psi}(\vec{x}),\partial_{k}S(\vec{X},t)\\}=\frac{p_{k}}{m\psi(\vec{x},t)}\frac{\partial}{\partial
x^{k}}\delta^{3}(\vec{x}-\vec{X}).$ (26)

This term can be eliminated by imposing the constraint $p_{k}\approx 0$. For
consistency we require that $\dot{p}_{k}=\\{p_{k},H\\}\approx 0$. From
equation (22) we see that this is satisfied, so that no new constraints arise.
The Hamiltonian $H$ plus the constraint $p_{k}\approx 0$ therefore generates
Schrodinger’s equation and the equation of motion for the particle.

The generalization of (23) to a system of $N$ particles is straightforward.
The Hamiltonian is given by

$H=\frac{i}{\hbar}\int\left[\Pi_{\psi}\left(\sum_{k=1}^{N}\frac{\hbar^{2}}{2m_{k}}\nabla^{2}_{k}\psi-V\psi\right)-\Pi_{\psi^{*}}\left(\sum_{k=1}^{N}\frac{\hbar^{2}}{2m_{k}}\nabla^{2}_{k}\psi^{*}-V\psi^{*}\right)\right]d^{3}x_{1}...d^{3}x_{N}+\sum_{k=1}^{N}\frac{\vec{p}_{k}}{m_{k}}\cdot\vec{\nabla}_{k}S\;,$
(27)

where $\vec{\nabla}_{k}S$ can also be written as

$\vec{\nabla}_{k}S=\sum_{k=1}^{N}\frac{\hbar}{2i|\psi|^{2}}\left[\psi^{*}\vec{\nabla}_{k}\psi-\psi\vec{\nabla}_{k}\psi^{*}\right]\;,$
(28)

$\psi=\psi(\vec{x}_{1}...,\vec{x}_{N})$ and
$V=V(\vec{x}_{1}...,\vec{x}_{N},t)$. This Hamiltonian plus the constraints
$\vec{p}_{k}\approx 0$ gives both the field and particle equations of motion.

A Hamiltonian formulation of the pilot-wave theory has also been developed by
Holland [9]. In his approach the wave function is written as
$\psi=\sqrt{\rho}e^{iS/\hbar}$ and the Hamiltonian is given by

$H_{tot}=H+\int\left\\{-\pi_{\rho}\left[\frac{1}{m}\frac{\partial}{\partial
q_{i}^{\prime}}\left(\rho\frac{\partial S}{\partial
q_{i}^{\prime}}\right)\right]-\pi_{S}\left[\frac{1}{2m}\left(\frac{\partial
S}{\partial q_{i}^{\prime}}\right)\left(\frac{\partial S}{\partial
q_{i}^{\prime}}\right)+Q+V\right]\right\\}d^{3}q^{\prime},$ (29)

where

$H=\sum_{i}\frac{p_{i}^{2}}{2m}+V+Q\;,$ (30)

$p_{i}$ is the momentum of the $i^{th}$ particle, V is the classical potential
and $Q$ is the quantum potential which is given by

$Q=-\frac{\hbar^{2}}{2m\sqrt{\rho}}\frac{\partial^{2}\sqrt{\rho}}{\partial
q_{i}^{2}}\;.$ (31)

Hamilton’s equations for $\dot{S}$ and $\dot{\rho}$ give Schrodinger’s
equation and Hamilton’s equations for the particle degrees of freedom are
given by

$\dot{q}_{i}=p_{i}/m\;\;\;\;\;\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;\;\;\dot{p}_{i}=-\frac{\partial}{\partial
q_{i}}[V+Q]\;.$ (32)

Combining these two equations gives

$m\ddot{q}_{i}=-\frac{\partial}{\partial q_{i}}[V+Q]\;,$ (33)

which is the second order form of the theory. This equation can be derived
from the first order guidance equation, but it is not equivalent to the
guidance equation. Holland then considers Liouville’s equation and shows, for
pure states, that the momentum is constrained to satisfy the guidance
equation. This approach differs significantly from the approach taken in this
paper.

##  3 Relativistic Pilot-Wave Theory

In this section I construct a Hamiltonian that gives the Dirac equation plus
the guidance equation.

The Lagrangian for the Dirac field coupled to the electromagnetic field is
given by

$L=\frac{i}{2}\left[\bar{\psi}\gamma^{\mu}D_{\mu}\psi-(D^{*}_{\mu}\bar{\psi)}\gamma^{\mu}\psi\right]-m\bar{\psi}\psi\;,$
(34)

where $D_{\mu}=\partial_{\mu}+ieA_{\mu}$ and I have taken $\hbar=c=1$. There
are two primary constraints

$\phi_{1}=\Pi_{\psi}-\frac{1}{2}i\bar{\psi}\gamma^{0}\approx 0\hskip
28.90755ptand\hskip
28.90755pt\phi_{2}=\Pi_{\bar{\psi}}+\frac{1}{2}i\gamma^{0}\psi\approx 0$ (35)

that satisfy

$\\{\phi_{1_{k}}(\vec{x}),\phi_{1_{\ell}}(\vec{y})\\}=\\{\phi_{2_{k}}(\vec{x}),\phi_{2_{\ell}}(\vec{y})\\}=0$
(36)

and

$\\{\phi_{1_{k}}(\vec{x}),\phi_{2_{\ell}}(\vec{y})\\}=-i(\gamma^{0})_{\ell
k}\delta^{3}(\vec{x}-\vec{y})\;.$ (37)

The constraints are, therefore, second class. Following the same procedure
that was used for Schrodinger’s equation gives

$H_{T}=i\int\left\\{\Pi_{\psi}\gamma^{0}\left[i\gamma^{k}\partial_{k}\psi-
eA_{\mu}\gamma^{\mu}\psi-m\psi\right]+\left[i(\partial_{k}\bar{\psi})\gamma^{k}+e\bar{\psi}A_{\mu}\gamma^{\mu}+m\bar{\psi}\right]\gamma^{0}\Pi_{\bar{\psi}}\right\\}d^{3}x.$
(38)

In the pilot-wave theory the velocity of the particle is given by [13, 14]

$\frac{dX^{\mu}}{d\tau}=J^{\mu}$ (39)

where

$J^{\mu}=\frac{\bar{\psi}\gamma^{\mu}\psi}{\sqrt{a^{2}+b^{2}}}\;,\;\;\;\;\;\;\;\;\;\;J^{\mu}J_{\mu}=1\;,$
(40)
$a=\bar{\psi}\psi\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;b=i\bar{\psi}\gamma^{5}\psi\;.$
(41)

The particle Hamiltonian is

$H_{P}=p_{\mu}J^{\mu}$ (42)

and the Hamiltonian for the field and particle is given by

$H=H_{T}+H_{p}\;.$ (43)

As in the non-relativistic approach we must impose $p_{\mu}\approx 0$ so that
addition of $H_{p}$ to $H_{T}$ does not alter the field equations for
$\Pi_{\psi}$ and $\Pi_{\bar{\psi}}$.

##  4 Pilot-Wave Approach in Scalar Quantum Field Theory

In this section I consider a massive scalar field and find a Hamiltonian that
gives Schrodinger’s equation plus the guidance equation for the field (see
[15, 16] for discussions on the pilot-wave theory for scalar fields).

Consider a real scalar field $\phi(x^{\mu})$ with the Lagrangian

$L=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}\;.$
(44)

The canonical momentum is given by

$\Pi=\frac{\partial\phi}{\partial t}$ (45)

and the Hamiltonian is given by

$H=\frac{1}{2}\int\left[\Pi^{2}+(\nabla\phi)^{2}+m^{2}\phi^{2}\right]d^{3}x\;.$
(46)

Schrodinger’s equation for this theory is given by

$i\frac{\partial\Psi}{\partial
t}=\frac{1}{2}\int\left[-\frac{\delta^{2}}{\delta\phi^{2}}+(\nabla\phi)^{2}+m^{2}\phi^{2}\right]d^{3}x\;\Psi\;.$
(47)

The wave function can be written as

$\Psi=Re^{iS}$ (48)

and the equation of motion is taken to be

$\Pi=\frac{\partial\phi}{\partial t}=\frac{\delta S}{\delta\phi}\;.$ (49)

Consider confining the field to a box of length $L$ with periodic boundary
conditions. The field can be written as

$\phi(\vec{x},t)=\sqrt{\frac{1}{V}}\sum_{\bf{k}}q_{\bf{k}}(t)e^{i\bf{k}\cdot\bf{x}}\;,$
(50)

where

${\bf{k}}=\frac{2\pi}{L}(n_{x},n_{y},n_{z})\;,$ (51)

$n_{x},n_{y}$ and $n_{z}$ integers, $q_{\bf{k}}^{*}=q_{-\bf{k}}$ and
$p_{\bf{k}}^{*}=p_{-\bf{k}}$. Schrodinger’s equation is given by

$i\frac{\partial\Psi}{\partial
t}=\sum_{{\bf{k}}/2}\left[-\frac{\partial^{2}}{\partial q_{\bf{k}}\partial
q^{*}_{\bf{k}}}+(k^{2}+m^{2})q_{\bf{k}}q^{*}_{\bf{k}}\right]\Psi\;,$ (52)

where $\Psi=\Psi(q_{\bf{k}},q^{*}_{\bf{k}},t)$ and ${\bf{k}}/2$ indicates that
the sum is carried out over half of the values of ${\bf{k}}$.

A Lagrangian that produces (52) is

$L=\frac{1}{2}i\left[\Psi^{*}\frac{\partial\Psi}{\partial
t}-\Psi\frac{\partial\Psi^{*}}{\partial
t}\right]-\sum_{{\bf{k}}/2}\left[\frac{\partial{\Psi}}{{\partial
q_{\bf{k}}}}\frac{\partial{\Psi^{*}}}{{\partial
q^{*}_{\bf{k}}}}+(k^{2}+m^{2})q_{\bf{k}}q^{*}_{\bf{k}}\Psi\Psi^{*}\right]\;.$
(53)

Following the same procedure that was discussed in section 2 leads to the
constraints

$\Pi_{\Psi}\approx\frac{1}{2}i\hbar\Psi^{*}\hskip 28.90755pt\hskip
28.90755pt\Pi_{\Psi^{*}}\approx-\frac{1}{2}i\hbar\Psi$ (54)

and to the total Hamiltonian

$H_{T}=i\sum_{{\bf{k}}/2}\int\left\\{\Pi_{\Psi}\left[\frac{\partial^{2}\Psi}{\partial
q_{\bf{k}}\partial
q^{*}_{\bf{k}}}-(k^{2}+m^{2})q_{\bf{k}}q^{*}_{\bf{k}}\Psi\right]-\Pi_{\Psi^{*}}\left[\frac{\partial^{2}\Psi^{*}}{\partial
q_{\bf{k}}\partial
q^{*}_{\bf{k}}}-(k^{2}+m^{2})q_{\bf{k}}q^{*}_{\bf{k}}\Psi^{*}\right]\right\\}dqdq^{*}\;,$
(55)

where $dq=\Pi_{{\bf{k/2}}}dq_{{\bf{k}}}$ and
$dq^{*}=\Pi_{{\bf{k/2}}}dq^{*}_{{\bf{k}}}$. This Hamiltonian gives the correct
equations of motion assuming Gauss’ theorem (and the vanishing of surface
terms) in infinite dimensional spaces. The guidance equations are given by

$\dot{q}_{\bf{k}}=\frac{\partial S}{\partial
q^{*}_{\bf{k}}}\;\;\;\;\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;\dot{q}^{*}_{\bf{k}}=\frac{\partial
S}{\partial q_{\bf{k}}}\;.$ (56)

This equation of motion follows from the Hamiltonian

$H_{p}=\sum_{{\bf{k}}/2}\left[p_{\bf{k}}\frac{\partial S}{\partial
q^{*}_{\bf{k}}}+p^{*}_{\bf{k}}\frac{\partial S}{\partial
q_{\bf{k}}}\right]\;,$ (57)

where

$\\{q_{\bf{k}},p_{\bf{l}}\\}=\\{q^{*}_{\bf{k}},p^{*}_{\bf{l}}\\}=\delta_{\bf{kl}}$
(58)

and all other Poisson brackets vanish. The equations of motion of the particle
and field follow from the Hamiltonian

$H=H_{T}+H_{p}$ (59)

plus the constraints $p_{\bf{k}}\approx 0$ and $p^{*}_{\bf{k}}\approx 0$.

##  5 Conclusion

In this paper I showed that the Hamiltonian

$H=\frac{i}{\hbar}\int\left[\Pi_{\psi}\left(\sum_{k=1}^{N}\frac{\hbar^{2}}{2m_{k}}\nabla^{2}_{k}\psi-V\psi\right)-\Pi_{\psi^{*}}\left(\sum_{k=1}^{N}\frac{\hbar^{2}}{2m_{k}}\nabla^{2}_{k}\psi^{*}-V\psi^{*}\right)\right]d^{3}x_{1}...d^{3}x_{N}+\sum_{k=1}^{N}\vec{p}_{k}\cdot\vec{\nabla}_{k}S$
(60)

plus the constraints $\vec{p}_{k}\approx 0$ gives Schrodinger’s equation for a
collection of non-relativistic particle in a potential and the guidance
equation

$\frac{d\vec{X}_{k}}{dt}=\vec{\nabla}_{k}S(\vec{X},t)\;,$ (61)

where $\vec{X}_{k}$ is the position of the $k^{th}$ particle. It is important
to note that the canonical momenta $\vec{p}_{k}$ are not related to
$\vec{v}_{k}$ and it is consistent to impose the constraints
$\vec{p}_{k}\approx 0$.

I then showed that

$H=i\int\left\\{\Pi_{\psi}\gamma^{0}\left[i\gamma^{k}\partial_{k}\psi-
eA_{\mu}\gamma^{\mu}\psi-m\psi\right]+\left[i(\partial_{k}\bar{\psi})\gamma^{k}+e\bar{\psi}A_{\mu}\gamma^{\mu}+m\bar{\psi}\right]\gamma^{0}\Pi_{\bar{\psi}}\right\\}d^{3}x+p_{\mu}J^{\mu},$
(62)

where

$J^{\mu}=\frac{\bar{\psi}\gamma^{\mu}\psi}{\sqrt{a^{2}+b^{2}}}\;,$ (63)
$a=\bar{\psi}\psi\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;b=i\bar{\psi}\gamma^{5}\psi$
(64)

plus the constraint $p_{\mu}\approx 0$ gives the Dirac equation and the
guidance equation

$\frac{dX^{\mu}}{d\tau}=J^{\mu}\;.$ (65)

Lastly I considered the massive Klein-Gordon equation as a quantum field (not
as a single particle wave equation). After decomposing the field into normal
modes

$\phi(\vec{x},t)=\sqrt{\frac{1}{V}}\sum_{\bf{k}}q_{\bf{k}}(t)e^{i\bf{k}\cdot\bf{x}}$
(66)

I showed that the Hamiltonian

$\begin{array}[]{cc}H=i\sum_{{\bf{k}}/2}\int\left\\{\Pi_{\Psi}\left[\frac{\partial^{2}\Psi}{\partial
q_{\bf{k}}\partial
q^{*}_{\bf{k}}}-(k^{2}+m^{2})q_{\bf{k}}q^{*}_{\bf{k}}\Psi\right]-\Pi_{\Psi^{*}}\left[\frac{\partial^{2}\Psi^{*}}{\partial
q_{\bf{k}}\partial
q^{*}_{\bf{k}}}-(k^{2}+m^{2})q_{\bf{k}}q^{*}_{\bf{k}}\Psi^{*}\right]\right\\}dqdq^{*}&\\\
&\\\ +\sum_{{\bf{k}}/2}\left[p_{\bf{k}}\frac{\partial S}{\partial
q^{*}_{\bf{k}}}+p^{*}_{\bf{k}}\frac{\partial S}{\partial
q_{\bf{k}}}\right]\;,\end{array}$ (67)

where $dq=\Pi_{{\bf{k/2}}}dq_{{\bf{k}}}$ and
$dq^{*}=\Pi_{{\bf{k/2}}}dq^{*}_{{\bf{k}}}$, plus the constraints
$p_{\bf{k}}\approx 0$ and $p^{*}_{\bf{k}}\approx 0$ gives Schrodinger’s
equation and the guidance equations

$\dot{q}_{\bf{k}}=\frac{\partial S}{\partial
q^{*}_{\bf{k}}}\;\;\;\;\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;\dot{q}^{*}_{\bf{k}}=\frac{\partial
S}{\partial q_{\bf{k}}}\;.$ (68)

## Acknowledgements

This research was supported by the Natural Sciences and Engineering Research
Council of Canada.

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