# Highly covariant quantum lattice gas model of the Dirac equation

Jeffrey Yepez  Air Force Research Laboratory, 29 Randolph Road, Hanscom AFB,
Massachusetts 01731, USA

(November 17, 2010)

###### Abstract

We revisit the quantum lattice gas model of a spinor quantum field theory—the
smallest scale particle dynamics is partitioned into unitary collide and
stream operations. The construction is covariant (on all scales down to a
small length $\ell$ and small time $\tau=c\,\ell$) with respect to Lorentz
transformations. The mass $m$ and momentum $p$ of the modeled Dirac particle
depend on $\ell$ according to newfound relations
$m=m_{\circ}\cos\frac{2\pi\ell}{\lambda}$ and
$p=\frac{\hbar}{\ell}\sin\frac{2\pi\ell}{\lambda}$, respectively, where
$\lambda$ is the Compton wavelength of the modeled particle. These relations
represent departures from a relativistically invariant mass and the de Broglie
relation—when taken as quantifying numerical errors the model is physically
accurate when $\ell\ll\lambda$. Calculating the vacuum energy in the special
case of a massless spinor field, we find that it vanishes (or can have a small
positive value) for a sufficiently large wave number cutoff. This is a marked
departure from the usual behavior of such a massless field.

quantum computation, quantum lattice gas, Dirac equation, discrete particle
dynamics, quantum field theory model, vacuum energy density, cosmological
constant

###### pacs:

03.67.Lx, 03.65.Pm, 04.25.Dm, 98.80.Es

††preprint:  v1.0

##  I Introduction

We consider the dynamics of a spinor quantum field where spacetime becomes
discrete at scales smaller than some fundamental length. In particular, we
revisit the quantum computational lattice representation known as the quantum
lattice gas model, a dynamical Feynman chessboard model of the Dirac equation
Feynman (1946); Feynman and Hibbs (1965). Variations, rediscoveries and
improvements of the Feynman chessboard model have appeared over the years,
including a model in 3+1 dimensions by Riazanov Riazanov (June 1958), an Ising
spin chain representation by Jacobson and Schulman Jacobson (1984); Jacobson
and Schulman (1984), a fundamental deterministic model by ’t Hooft ’t Hooft
(1988, 1997), a lattice Boltzmann model by Succi and Benzi quantum Succi and
Benzi (1993), a unitary model by Bialynicki-Birula Bialynicki-Birula (1994),
and quantum lattice gas models in 1+1 dimensions by Meyer Meyer (1996) and in
3+1 dimensions by this author Yepez (2005). We presently consider a
representation that retains 4-momentum conservation
$E^{2}=(cp)^{2}+(mc^{2})^{2}$ of special relativity down to a small length
scale $\ell$ and time scale $\tau$. The low-energy limit of the lattice model
is defined as the dynamical regime where the Compton wavelength $\lambda$ of
the quantum particle represented by an amplitude field $\psi(x)$ is much
larger than the small scale. $\psi(x)$ is treated as continuous for
$\lambda\ggg\ell$. Continuous derivatives emerge as effective quantum
operators and the particle physics may be represented by the Dirac Lagrangian
${\cal L}_{\text{\tiny
Dirac}}=\overline{\psi}(i\gamma^{\mu}\partial_{\mu}-m_{\circ})\psi$, where the
Dirac matrices are $\gamma^{\mu}=(\gamma^{0},\gamma^{i})$, the spacetime
derivatives are $\partial_{\mu}=(\partial_{t},\partial_{i})$ for $i=1,2,3$,
and $m_{\circ}$ is the “invariant” particle mass (here expressed in natural
units with $\hbar=1$ and $c=1$ for convenience).

This paper is organized as follows. We begin in Sec. II by formally
introducing the quantum lattice gas model as a Langanian based theory. Then,
in Sec. III, we present a mapping procedure whereby the discrete dynamics of a
quantum lattice gas model is made equivalent to the Dirac equation. This
procedure leads to analytical form of the particle momentum that is a
modification of the de Broglie relation. In Sec. IV, we present a deviation of
the quantum lattice gas stream and collide operators that form the basis of
our quantum algorithm for the Dirac equation. In particular, we derive a
unitary expression for the collision operator that is serves as a mechanism to
give the spinor field its mass. Our primary intent is to show that the quantum
lattice gas, taken as a numerical tool for this quantum computational physics
application, provides a high degree of numerical accuracy. Then, in Sec. V, we
examine the newfound requirements to have the dynamical equation of motion of
the quantum lattice gas model equal the Dirac equation at a selected small
scale and explore the consequences of these requirements. We present a
calculation of the vacuum energy density of a spinor field, treating the
special case of a massless spinor field. Following a detailed analysis of the
behavior the error terms, one finds an alternate theoretical purpose of the
quantum lattice gas as a toy model. It provides an example where the vacuum
energy of a massless spinor field can vanish or be very small. That is, one
can take the quantum lattice gas as a toy model of Planckian scale physics and
thus set the small scale sizes $\ell$ and $\tau$ to the Planck length
$\ell_{\text{\tiny P}}=\sqrt{\hbar G/c^{3}}$ and Planck time
$\tau_{\text{\tiny P}}=\ell/c$, providing a route for a small positive
cosmological constant. In Sec. VI is a conclusion and summary.

##  II Quantum lattice gas model

The proposed high-energy (small scale) quantum lattice gas representation may
be formally expressed by the Lagrangian density of the form

${\cal L}^{\text{\tiny
grid}}\\!\\!=\overline{\psi}\left[i\gamma^{0}\frac{e^{\tau\partial_{t}}-e^{-\tau(\gamma^{0}\cdot\gamma^{i}\partial_{i}+im\gamma^{0})}}{\tau}\right]\psi={\cal
L}_{\text{\tiny Dirac}}+{\cal O}(\tau^{2}).$ (1a) By the least action
principle, this Lagrangian density leads to the equation of motion of the form
$e^{\tau\partial_{t}}\psi(x)=e^{-\tau\gamma^{0}\cdot\gamma^{i}\partial_{i}}e^{-i\tau
m\gamma^{0}}\psi(x).$ (1b) Equation (1b) is the equation of motion of a
quantum lattice gas, a unitary model for a system of noninteracting Dirac
particles. On the right-hand side of (1b), free chiral particle motion is
given by a stream operator $U_{\text{\tiny
S}}=e^{i\tau\gamma^{0}\cdot\gamma^{i}p_{i}}$, with momentum operator
$p_{i}=-i\partial_{i}$. A mass-generating interaction that causes a lefthanded
particle to flip into a right-handed particle (and vice versa) is given by a
unitary collision operator $U_{\text{\tiny C}}=e^{-i\tau m\gamma^{0}}$. The
product $U_{\text{\tiny QLG}}=U_{\text{\tiny S}}U_{\text{\tiny C}}$ is the
local evolution operator of a quantum lattice gas system acting on the spinor
field $\psi^{\text{\tiny T}}(x)={\begin{pmatrix}\psi_{\text{\tiny
L}}(x)&\psi_{\text{\tiny R}}(x)\end{pmatrix}}$. The lefthand side of (1b) is a
newly computed state $\psi^{\prime}(x)\equiv e^{\tau\partial_{t}}\psi(x)$ at
time $t+\tau$, so (1b) may be written as a quantum algorithmic map
$\psi^{\prime}(x)=U_{\text{\tiny S}}U_{\text{\tiny C}}\psi(x)\mapsto\psi(x),$
(1c)

taken to be homogeneously applied at all points $\bm{x}$ of space and at all
increments $t$ of time. In natural units ($\hbar=1$ and $c=1$), the quantum
lattice gas model (1c) is specified in $1+1$ dimensions by the following
unitary operators:

$\displaystyle U_{\text{\tiny S}}^{z}$ $\displaystyle=$ $\displaystyle
e^{i\ell p_{z}\sigma_{z}}$ (2a) $\displaystyle U_{\text{\tiny C}}$
$\displaystyle=$
$\displaystyle\sqrt{1-m_{\circ}^{2}\tau^{2}}-i\sigma_{x}e^{i\sigma_{z}\,\ell
p_{z}}\,m_{\circ}\tau,$ (2b)

where $m_{\circ}$ is the mass of the modeled Dirac particle in the low-energy
limit $\ell/\lambda\sim 0$.

In the low-energy limit, the ${\cal O}(\tau^{2})$ error terms on the righthand
side of (1a) become negligible, so ${\cal L}^{\text{\tiny grid}}\sim{\cal
L}_{\text{\tiny Dirac}}$ is covariant with respect to Lorentz transformations
in this limit. Yet, can ${\cal L}^{\text{\tiny grid}}$ be manifestly covariant
at high-energies $\ell/\lambda\sim 1$? We consider how to achieve the
covariance of (1a) at a small scale: it necessarily occurs when the high-
energy equation (1b)—or equivalently the quantum lattice gas equation (1c)—has
the form of the Dirac equation $(\gamma^{\mu}p_{\mu}+m)\psi(x)=0$.

The model is an expression of the simple idea of a spacetime manifold that
becomes discrete below a small scale $\ell$. The prescriptions needed to make
(1c) equivalent to the Dirac equation are derived in the following section.

##  III Imposing covariance at the small scale

Here we show that (1c) in the high-energy limit can be made equivalent to the
Dirac equation. We begin with a local evolution operator as a composition of
“qubit rotations”
$U_{\hat{\bm{n}}_{2}}=e^{-i\frac{\beta_{2}}{2}\hat{\bm{n}}_{2}\cdot\bm{\sigma}}$
and
$U_{\hat{\bm{n}}_{1}}=e^{-i\frac{\beta_{1}}{2}\hat{\bm{n}}_{1}\cdot\bm{\sigma}}$:

$\displaystyle
U_{\hat{\bm{n}}_{2}}(\beta_{2})U_{\hat{\bm{n}}_{1}}(\beta_{1})\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!e^{-i\frac{\beta_{2}}{2}\hat{\bm{n}}_{2}\cdot\bm{\sigma}}e^{-i\frac{\beta_{1}}{2}\hat{\bm{n}}_{1}\cdot\bm{\sigma}}$
(3b) $\displaystyle=$
$\displaystyle\cos\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}-\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}\hat{\bm{n}}_{1}\cdot\hat{\bm{n}}_{2}$
$\displaystyle-$ $\displaystyle
i\Big{[}\sin\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}\hat{\bm{n}}_{1}+\cos\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}\hat{\bm{n}}_{2}$
$\displaystyle-\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}\hat{\bm{n}}_{1}\times\hat{\bm{n}}_{2}\Big{]}\cdot\bm{\sigma},$

where $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is a vector of Pauli
matrices, $\hat{\bm{n}}_{1}$ and $\hat{\bm{n}}_{2}$ are unit vectors
specifying the respective principal axes of rotation, and $\beta_{1}$ and
$\beta_{2}$ are real-valued rotation angles.111In (3b) we used the identity
$(\bm{a}\cdot\bm{\sigma})\cdot(\bm{b}\cdot\bm{\sigma})=\bm{a}\cdot\bm{b}+i\,\left(\bm{a}\times\bm{b}\right)\cdot\bm{\sigma}$.
Let us take $U_{\text{\tiny
S}}^{z}=e^{-i\frac{\beta_{2}}{2}\hat{\bm{n}}_{2}\cdot\bm{\sigma}}$ as our
stream operator and $U_{\text{\tiny
C}}=e^{-i\frac{\beta_{1}}{2}\hat{\bm{n}}_{1}\cdot\bm{\sigma}}$ as our
collision operator. Without loss of generality, we may choose the principle
axis of rotation along the $\hat{\bm{z}}$ to generate streaming,

$U_{\text{\tiny S}}^{z}=e^{i\ell
p_{z}\sigma_{z}/\hbar}=e^{-i\frac{\beta_{2}}{2}\sigma_{z}},$ (4a) and treat
the quantum algorithmic map as if it were applied in 1+1 dimensions.222 The
reduction from 3+1 to 1+1 dimensions is allowed because the algorithm has the
product form $\psi^{\prime}(x)=U_{\text{\tiny S}}U_{\text{\tiny
C}}\psi(x)\mapsto\psi(x),$ where $U_{\text{\tiny
S}}=e^{-i\frac{\pi}{4}\sigma_{y}}U_{\text{\tiny
S}}^{x}e^{i\frac{\pi}{4}(\sigma_{y}+\sigma_{x})}U_{\text{\tiny
S}}^{y}e^{-i\frac{\pi}{4}\sigma_{x}}U_{\text{\tiny
S}}^{z}=e^{i\tau\gamma^{0}\cdot\gamma^{i}p_{i}}$, with Dirac matrices
$\gamma^{0}=\sigma_{x}\otimes\bm{1}$ and
$\gamma^{i}=i\sigma_{y}\otimes\sigma_{i}$ in the chiral representation Succi
and Benzi (1993); Yepez (2005). Streaming in each of the spatial directions
occurs independently, so for simplicity we can choose to consider a Dirac wave
moving along $\hat{\bm{z}}$.  In this frame a general collision operator is
$U_{\text{\tiny
C}}=e^{-i\frac{\beta_{1}}{2}(\alpha\sigma_{x}+\beta\sigma_{y}+\gamma\sigma_{z})},$
(4b)

where $\alpha$, $\beta$, and $\gamma$ are real valued components subject to
the constraint $\alpha^{2}+\beta^{2}+\gamma^{2}=1$. The unitary operators (4)
are applied locally and homogeneously at all the points in the system. That
is, we consider a construction whereby the two principal unit vectors
specifying the axes of rotation are

$\displaystyle\hat{\bm{n}}_{1}$ $\displaystyle=$
$\displaystyle(\alpha,\beta,\gamma)\qquad\qquad\hat{\bm{n}}_{2}=(0,0,1).$ (5)

With this choice, $\hat{\bm{n}}_{1}\times\hat{\bm{n}}_{2}=(\beta,-\alpha,0)$
and $\hat{\bm{n}}_{1}\cdot\hat{\bm{n}}_{2}=\gamma$, so (3) is a quite general
representation of a quantum lattice gas evolution operator

$\displaystyle U_{\text{\tiny S}}^{z}\,U_{\text{\tiny C}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{axes_of_2_qubit_rotations})}}{{=}}$
$\displaystyle\cos\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}-\gamma\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}$
$\displaystyle-$ $\displaystyle
i\left(\alpha\sin\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}-\beta\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}\right)\sigma_{x}$
$\displaystyle-$ $\displaystyle
i\left(\beta\sin\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}+\alpha\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}\right)\sigma_{y}$
$\displaystyle-$ $\displaystyle
i\left(\gamma\sin\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}+\cos\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}\right)\sigma_{z}.$

The Dirac equation for a relativistic quantum particle of mass $m_{\circ}$ may
be written as

$i\hbar\partial_{t}\psi=-c\,p_{z}\sigma_{z}\psi+m_{\circ}c^{2}\sigma_{x}\psi.$
(7)

Its time-difference form may be written as

$\psi^{\prime}=\left(1+\frac{ic\,p_{z}\tau}{\hbar}\sigma_{z}-\frac{im_{\circ}c^{2}\tau}{\hbar}\sigma_{x}\right)\psi,$
(8)

for small $\tau$ and for momentum operator $p_{z}=-i\hbar\partial_{z}$. We may
view the unitary operator acting on the right-hand side of (8) as the
effective low-energy operator obtained from the quantum lattice gas operator
(III)

$U_{\text{\tiny S}}^{z}\,U_{\text{\tiny
C}}\xrightarrow{\text{small}~{}\ell}1+\frac{ic\,p_{z}\tau}{\hbar}\sigma_{z}-\frac{im_{\circ}c^{2}\tau}{\hbar}\sigma_{x}.$
(9)

To establish a correspondence between (III) and (9), we simply choose the
real-valued components of $\hat{\bm{n}}_{1}$ to satisfy the following three
conditions:

$\displaystyle\alpha\sin\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}$
$\displaystyle-$
$\displaystyle\beta\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}=\frac{m_{\circ}c^{2}\tau}{\hbar}$
(10a) $\displaystyle\beta\sin\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}$
$\displaystyle+$
$\displaystyle\alpha\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}=0$ (10b)
$\displaystyle\gamma\sin\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}$
$\displaystyle+$ $\displaystyle\ \
\cos\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}=-\frac{c\,p_{z}\tau}{\hbar}.$
(10c) Additionally, we should respect the reality condition that
$\hat{\bm{n}}_{1}$ have unit norm333 Alternatively, instead of (10d), we could
impose the condition that
$\cos\frac{\beta_{1}}{2}\cos\frac{\beta_{2}}{2}-\gamma\sin\frac{\beta_{1}}{2}\sin\frac{\beta_{2}}{2}=1,$
forcing (III) to be identical to (9). However, in this case, the resulting
solution for components of $\hat{\bm{n}}_{1}$ has $\alpha$ imaginary, and this
breaks the unitarity of $U_{\text{\tiny C}}$. So, we impose (10d) to strictly
enforce unitarity. $\alpha^{2}+\beta^{2}+\gamma^{2}=1$ (10d)

that we established above with the collision operator (4b). For the sake of
simplicity, let us start with a specialized construction whereby
$\hat{\bm{n}}_{1}$ is perpendicular to $\hat{\bm{n}}_{2}$. The solution of
(10) in this special case is

$\alpha=\cos\frac{\beta_{2}}{2}\qquad\beta=-\sin\frac{\beta_{2}}{2}\qquad\gamma=0.$
(11)

Inserting (11) into (10a) gives
$\sin\frac{\beta_{1}}{2}=\frac{m_{\circ}c^{2}\tau}{\hbar},$ and in turn (10c)
is
$\sqrt{1-\left(\frac{m_{\circ}c^{2}\tau}{\hbar}\right)^{2}}\sin\frac{\beta_{2}}{2}=-\frac{c\,p_{z}\tau}{\hbar}.$
In (4a) we chose $-\ell\,p_{z}/\hbar={\beta_{2}}/{2}$, so in turn we have

$\sqrt{1-\left(\frac{m_{\circ}c^{2}\tau}{\hbar}\right)^{2}}\sin\frac{\ell
p_{z}}{\hbar}=\frac{c\,p_{z}\tau}{\hbar}.$ (12)

This is a grid equation that relates the cell sizes $\ell$ and $\tau$ to the
mass and momentum of the quantum particle in an intrinsic way. Equation (12)
can be interpreted as a rather fundamental relativistic relationship between
particles and points. In place of the theory of special relativity for
classical particle dynamics in a continuum, here we have constructed a
lattice-based version of special relativity for particle dynamics emerging at
a small scale where the spacetime foam has a regular structure.

Let us consider some implications of (12). Squaring (12) gives

$\left(\frac{\hbar}{\tau}\sin\frac{\ell
p_{z}}{\hbar}\right)^{2}-\left(m_{\circ}c^{2}\sin\frac{\ell
p_{z}}{\hbar}\right)^{2}=(cp_{z})^{2}.$ (13a)

Then adding $m_{\circ}^{2}c^{4}$ to both sides, we have

$\left(\frac{\hbar}{\tau}\right)^{2}\sin^{2}\frac{\ell
p_{z}}{\hbar}+\left(m_{\circ}c^{2}\right)^{2}\cos^{2}\frac{\ell
p_{z}}{\hbar}=(cp_{z})^{2}+(m_{\circ}c^{2})^{2}.$ (13b)

This is a candidate grid-level relativistic energy equation that leads us to
define a grid momentum $p_{z}^{\text{\tiny grid}}$ and a grid mass $m$
dependent on $\ell$ as follows:

$\displaystyle p_{z}^{\text{\tiny grid}}$ $\displaystyle\equiv$
$\displaystyle\frac{\hbar}{c\tau}\sin\frac{\ell p_{z}}{\hbar}\qquad\qquad
m\equiv m_{\circ}\cos\frac{\ell p_{z}}{\hbar}.$ (14)

Hence, the lefthand side of (13b) can be interpreted as a redefinition of the
Dirac particle’s kinetic and rest energies. Inserting the de Broglie relation
($p_{z}=h/\lambda$ momentum eigenvalue), the grid mass and momentum become

$\displaystyle p_{z}^{\text{\tiny grid}}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{c\tau}\sin\frac{2\pi\ell}{\lambda}\qquad\qquad
m=m_{\circ}\,\cos\frac{2\pi\ell}{\lambda}.$ (15)

In the low-energy limit defined by $\lambda\ggg 2\pi\ell$, expanding (12) to
first order implies that the space and time cell sizes are linearly related by
the speed light $\ell=c\,\tau,$ an intuitive relationship that we expect to
hold. In the low-energy limit, (15) reduces to

$\displaystyle p_{z}$ $\displaystyle=$
$\displaystyle\frac{h}{\lambda}\qquad\qquad m=m_{\circ}.$ (16)

That is, the low-energy limit of (15) corresponds to a usual quantum particle
with an invariant mass that is entirely independent of the particle’s
momentum, and the quantum particle acts like a wave according to standard
quantum mechanics. However, there is a marked departure from standard quantum
mechanics in the high-energy limit in the region $\lambda\lesssim 20\ell$ as
shown in Fig. 1.

##  IV The algorithm in natural units

For expediency, let us now switch our dimensional convention to the natural
units, $\hbar=1$ and $c=1$.444In the natural units $\hbar=1$ and $c=1$, length
and time have like dimension of length (i.e. $[\ell]=[\tau]=L$) while mass,
momentum, and energy values have like dimension of inverse length (i.e.
$[m]=[p]=[E]=L^{-1}$). Any expression written in the natural units can be
converted back to an expression in the dimensionful $M,L,T$ units by simply
reinserting the speed of light and Planck’s constant by
$\ell\mapsto\frac{\ell}{c}$, $m\mapsto\frac{mc^{2}}{\hbar}$,
$p\mapsto\frac{pc}{\hbar}$, and $E\mapsto\frac{E}{\hbar}$. We can write (12)
as

$\displaystyle\sqrt{1-m_{\circ}^{2}\tau^{2}}\,\sin(\ell p_{z})$
$\displaystyle=$ $\displaystyle p_{z}\tau$ (17a)
$\displaystyle\sqrt{1-m_{\circ}^{2}\tau^{2}}\,\cos(\ell p_{z})$
$\displaystyle=$ $\displaystyle\sqrt{1-E^{2}\tau^{2}},$ (17b) or equivalently
as $e^{i\ell
p_{z}}=\exp\left[i\cos^{-1}\sqrt{\frac{1-E^{2}\tau^{2}}{1-m_{\circ}^{2}\tau^{2}}}\right].$

Furthermore, our solution (11) implies that the rotation axis for the
collision operator is $\hat{\bm{n}}_{1}=\hat{\bm{x}}\cos\ell
p_{z}+\hat{\bm{y}}\sin\ell p_{z},$ and in turn this implies that the hermitian
generator of (4b) is $\hat{\bm{n}}_{1}\cdot\bm{\sigma}=\sigma_{x}\cos\ell
p_{z}+\sigma_{y}\sin\ell p_{z}=\sigma_{x}e^{i\sigma_{z}\,\ell p_{z}}.$ Hence,
since $\sin\frac{\beta_{1}}{2}=m_{\circ}\tau$, we can explicitly represent the
collision operator $U_{\text{\tiny
C}}=e^{-i\frac{\beta_{1}}{2}\hat{\bm{n}}_{1}\cdot\bm{\sigma}}$ in terms of the
mass and momentum of the quantum particle as

$\displaystyle U_{\text{\tiny C}}$ $\displaystyle=$
$\displaystyle\sqrt{1-m_{\circ}^{2}\tau^{2}}-i\hat{\bm{n}}_{1}\cdot\bm{\sigma}\,m_{\circ}\tau$
(18a) $\displaystyle=$
$\displaystyle\sqrt{1-m_{\circ}^{2}\tau^{2}}-i\sigma_{x}e^{i\sigma_{z}\,\ell
p_{z}}\,m_{\circ}\tau.$ (18b)

Multiplying by the stream operator $U_{\text{\tiny S}}^{z}=e^{i\ell
p_{z}\sigma_{z}}$, the evolution operator (III) can now be explicitly
calculated

$\displaystyle U_{\text{\tiny S}}^{z}U_{\text{\tiny C}}\\!\\!\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!e^{i\ell
p_{z}\sigma_{z}}\sqrt{1-m_{\circ}^{2}\tau^{2}}-ie^{i\ell
p_{z}\sigma_{z}}\sigma_{x}e^{i\sigma_{z}\,\ell p_{z}}\,m_{\circ}\tau\qquad\ $
(19a) $\displaystyle=$ $\displaystyle e^{i\ell
p_{z}\sigma_{z}}\sqrt{1-m_{\circ}^{2}\tau^{2}}-i\sigma_{x}\,m_{\circ}\tau$
(19c) $\displaystyle=$ $\displaystyle\sqrt{1-m_{\circ}^{2}\tau^{2}}\cos
p_{z}\ell+i\sigma_{z}\sqrt{1-m_{\circ}^{2}\tau^{2}}\sin p_{z}\ell$
$\displaystyle-\ i\sigma_{x}\,m_{\circ}\tau$
$\displaystyle\stackrel{{\scriptstyle(\ref{grid_equation_natural_unit_component_form_a})}}{{\stackrel{{\scriptstyle(\ref{grid_equation_natural_unit_component_form_b})}}{{=}}}}$
$\displaystyle\sqrt{1-E^{2}\tau^{2}}+i\sigma_{z}p_{z}\tau-\
i\sigma_{x}\,m_{\circ}\tau$ (19d) $\displaystyle=$
$\displaystyle\sqrt{1-E^{2}\tau^{2}}+iE\tau\left(\sigma_{z}\frac{p_{z}}{E}-\sigma_{x}\,\frac{m_{\circ}}{E}\right).$
(19e)

This result leads us to define the rotation axis

$\hat{\bm{n}}_{12}\equiv-\frac{m_{\circ}}{E}\,\hat{\bm{x}}+\frac{p_{z}}{E}\,\hat{\bm{z}}.$
(20)

Since $(\hat{\bm{n}}_{12}\cdot\bm{\sigma})^{2}=1$ (an involution), we are free
to write (19e) in a manifestly unitary form
$e^{-i\frac{\beta_{12}}{2}\hat{\bm{n}}_{12}\cdot\bm{\sigma}}$ as follows:

$\displaystyle U_{\text{\tiny S}}^{z}U_{\text{\tiny C}}$ $\displaystyle=$
$\displaystyle\exp\left[i\cos^{-1}\\!\left(\sqrt{1-E^{2}\tau^{2}}\right)\hat{\bm{n}}_{12}\cdot\bm{\sigma}\right]\qquad$
(21a) $\displaystyle\stackrel{{\scriptstyle(\ref{n_12_m_p_E_form})}}{{=}}$
$\displaystyle\exp\left[i\,\frac{\cos^{-1}\sqrt{1-E^{2}\tau^{2}}}{E}\left(\sigma_{z}p_{z}-\sigma_{x}\,{m_{\circ}}\right)\right]\qquad$
(21b) $\displaystyle\cong$ $\displaystyle
e^{\left.-i\,\ell\middle(-\sigma_{z}p_{z}+\sigma_{x}\,{m_{\circ}}\right)},\qquad$
(21c)

where in the last line we made the identification

$\cos(E\ell)=\sqrt{1-E^{2}\tau^{2}},$ (22)

which is exact to one part in $10^{139}$ (i.e. accurate to 4th order in
$\ell$). This is equivalent to identifying the gate angle with the ratio of
the small scale length to the particle energy, $-{\beta_{12}/2}=\ell/E$. Then,
since $U_{\text{\tiny S}}^{z}U_{\text{\tiny C}}\equiv e^{-ih^{\text{\tiny
grid}}\tau}$, the high-energy Hamiltonian may be written as

$h^{\text{\tiny
grid}}\stackrel{{\scriptstyle(\ref{lattice_gas_operator_in_exp_form_c})}}{{=}}\left.\frac{\ell}{\tau}\middle(-p_{z}\sigma_{z}+m_{\circ}\sigma_{x}\right).$
(23)

Thus, we have successfully demonstrated that the quantum lattice gas equation
(1c) is equivalent to the Dirac equation as the hamiltonian generating its
unitary dynamics is the Dirac hamiltonian, even at the small scale.

##  V Consequences of the modified de Broglie relation

Imposing covariance on the high-energy representation (1) leads to two
departures (15), that we derived in the previous section, from the correct
behavior given by relativistic quantum field theory. First, the momentum of
the quantum particle must obey a small length $\ell$ dependent momentum
relation

$p=\frac{\hbar}{\ell}\sin\frac{2\pi\ell}{\lambda}$ (24a) in place of the de
Broglie relation $p=h/\lambda$. Second, the mass of the Dirac particle is no
longer taken as an invariant quantity—it must depend on the small length as
well as the particle’s Compton wavelength
$m=m_{\circ}\cos\frac{2\pi\ell}{\lambda},$ (24b)

where $m_{\circ}$ is a fixed constant, otherwise interpreted in the low-energy
limit as the invariant particle mass. Plots of (24) are given in Fig. 1 for
$m_{\circ}$ set to the electron mass. Notice that (24a) vanishes for
$\lambda=\ell$ and $\lambda=2\ell$ and oscillates about zero as
$\lambda\rightarrow 0$. Also notice that (24b) vanishes at $\lambda=4\ell$ and
is in fact negative for $\lambda=2\ell$ and $\lambda=3\ell$, again oscillating
as $\lambda\rightarrow 0$. These departures from standard quantum mechanics
and special relativity theory are rather consequential at very small scales,
departing on scales $\lesssim 20\ell$ that can be viewed as a region where the
errors in the lattice gas model are dominate. Yet, at the relevant larger
scales (far above the small scale $\ell$), the physics of the quantum lattice
gas model is indistinguishable from that predicted by the relativistic quantum
field theory representation of Dirac fields.

Figure 1:  Log-log plot of (24) for mass (red dots) and momentum (blue dots)
in GeV of a single proton versus its wavelength with the small scale set to
the Planck length, $\ell\equiv\ell_{\text{\tiny P}}=1.616\times 10^{-35}$m.
The straight lines are the de Broglie relation of quantum mechanics,
$p=h/\lambda$ (blue dashed line), and the invariant mass of special
relativity, $m_{\circ}=0.511$ MeV (red dashed line). Respectively, the slopes
are $-1$ and $0$ for the standard theories. The intersection of the mass and
momentum lines occurs at the Compton wavelength of the Dirac particle. The two
insets are linear plots in the extreme ultraviolet region,
$\ell/4\leq\lambda\leq 20\ell$, where the lattice theory departs from quantum
mechanics and special relativity.

Yet, in the context of the toy model, we should use (24) to calculate the
vacuum energy associated with a spin-$\frac{1}{2}$ fermion. When we integrate
over all space to determine the total density contained in a Dirac field we
have

$\displaystyle\rho_{\text{\tiny tot}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int\frac{d^{3}k}{c^{2}(2\pi)^{3}}\sqrt{(pc)^{2}+(mc^{2})^{2}}$
(25a)
$\displaystyle\stackrel{{\scriptstyle(\ref{discrete_relativity_premises})}}{{=}}$
$\displaystyle\frac{\hbar}{2\pi^{2}c\ell^{4}}\int_{0}^{k_{c}\ell}d(k\ell)({k\ell})^{2}\sqrt{\sin^{2}k\ell+\epsilon^{2}\cos^{2}k\ell},\qquad$
(25b)

where the quantity $\epsilon\equiv{m_{\circ}c\ell}/{\hbar}$ is small when
$m_{\circ}$ is much less than the mass,
$m_{\circ}\ll\hbar\,(\ell^{2}/\tau)^{-1}$. If we take $\epsilon=0$ to model a
massless relativistic particle, then performing the integration of (25b)
yields

$\displaystyle\rho_{\text{\tiny vac}}^{\text{\tiny theory}}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\left.\frac{\hbar}{2\pi^{2}c\ell^{4}}\left(2k\ell\sin
k\ell-(k^{2}\ell^{2}-2)\cos k\ell\right)\right|_{0}^{k_{c}\ell},\qquad$ (26)

which can be either positive, zero, or negative depending sensitively on the
value of the wave number cutoff $k_{c}$ as well as on the value of the small
scale $\ell$.

If one takes the quantum lattice gas as a simplistic representation of
Planckian scale physics (choosing $\ell=\ell_{\text{\tiny P}}$) where (24) is
interpreted as physical behavior instead of numerical grid error, then its
prediction of an allowable small value of the vacuum energy of a massless
spinor field may be viewed as a new physical mechanism. That is, if we use
(24) to calculate the vacuum energy associated with a spin-$\frac{1}{2}$
fermion, then the toy model can avoid the cosmological constant problem—it is
not necessarily $10^{121}$ times too large. Taking $k_{c}$ as a parameter in
(26), at $k_{c}=(4.08557...)/\ell$, we find that $\rho_{\text{\tiny
vac}}^{\text{\tiny theory}}=0$, as shown in Fig. 2. The experimentally
observed value, $\rho_{\text{\tiny vac}}^{\text{\tiny obs.}}=9.9\times
10^{-27}\text{kg}/\text{m}^{3}$ (such as obtained by the Wilkinson Microwave
Anisotropy Probe) is obtained at a slightly smaller wave number cutoff,
corresponding to a grid scale about four times smaller than $\ell$, a sub-
Planckian length scale.555Our simplistic estimate is further simplified by not
considering the inflationary epoch of space under an extremely high vacuum
energy density, just the dynamics of a fermionic field when the spacetime is
flat with a small positive cosmological constant when its discreteness below
the Planck scale becomes relevant.

Figure 2: Comparison of theoretical predictions of the vacuum energy density
as a function of wave number cutoff times the Planck length, $k_{c}\ell$. The
dashed curve is the standard quantum field theory prediction and the solid
curve is the quantum lattice gas prediction (26), where $\rho^{\text{\tiny
theory}}_{\text{\tiny vac}}=0$ at $k_{c}\,\ell=4.08557$. The theoretical
prediction at $k_{c}\ell\sim 1$ is $10^{121}$ times too large for both curves,
whereas for the solid curve $k_{c}\ell\gtrapprox 4$ is very close to the
experimentally observed value.

Alternatively, one can choose the Planck scale to be smaller than the grid
scale, $\ell_{\text{\tiny P}}\ll\ell\ll\lambda$. In this case, we still have
$\rho_{\text{\tiny vac}}^{\text{\tiny theory}}(4.08557k_{c}\ell)=0$. There
exists a real-valued number $C\lesssim 4.08557$ for which $\rho_{\text{\tiny
vac}}^{\text{\tiny obs.}}(Ck_{c}\ell)=\rho_{\text{\tiny vac}}^{\text{\tiny
obs.}}$, although we have not predicted this number and thus do not address
the fine-tunning problem. Considerations regarding an additional fundamental
length scale, in addition to the Planck scale, have recently appeared in Ref.
Klinkhamer (2007), including references therein.

##  VI Conclusion

We revisited the quantum lattice gas model with a unitary evolution operator
$U_{\text{\tiny S}}U_{\text{\tiny C}}$ applied at a small scale $\ell$ that
advances a Dirac spinor field, represented on a grid, forward by a small time
scale increment $\tau$. We derived the conditions for which the generator of
evolution is the Dirac Hamiltonian, $U_{\text{\tiny S}}U_{\text{\tiny C}}\cong
e^{\left.-i\,\ell\middle(-\sigma_{z}p_{z}+\sigma_{x}\,{m_{\circ}}\right)}$. We
quantified the error of the quantum lattice gas model as a departure from
standard quantum mechanical behavior for the particle momentum going as
$p=(\hbar/\ell)\sin(2\pi\ell/\lambda)$ and as its departure from a
relativistically invariant particle mass going as
$m=m_{\circ}\cos(2\pi\ell/\lambda)$. In this regard, the quantum lattice gas
model (2) is numerically accurate only to scales $\gtrsim 20\ell$, even though
it retains covariant behavior down for scales $\gtrsim\ell$. Yet, the
numerical error of the model can be taken as a good feature—providing a
mechanism for a small positive cosmological constant.

There have been a number of theoretical attempts employing, for example,
supersymmetry Nilles (1984), string theory Dine and Seiberg (1985), and the
anthropic principle Vilenkin (1998) to bridge the known chasm between the
large quantum field theory prediction of $\rho_{\text{\tiny vac}}^{\text{\tiny
qft}}\sim 10^{110}\text{eV}^{4}$ for the late-time, zero-temperature vacuum
energy density of “empty” space and the observed value of $\rho_{\text{\tiny
vac}}^{\text{\tiny obs.}}\sim 10^{-11}\text{eV}^{4}$ associated with a small
positive cosmological constant. The toy model presented herein gives a value
of $k_{c}$ for which the vacuum energy vanishes. This is the root of the
equation $2k_{c}\ell\sin k_{c}\ell-(k_{c}^{2}\ell^{2}-2)\cos k_{c}\ell-2=0$.
However, the theoretical considerations presented above do not tell us why the
wave number cutoff should be fined tuned so that $\rho_{\text{\tiny
vac}}^{\text{\tiny theory}}=\rho_{\text{\tiny vac}}^{\text{\tiny
obs.}}$.666There are several routes whereby a high-energy quantum lattice gas
model may predict a small positive vacuum energy density. At $k_{c}=2\pi/\ell$
we have $\rho_{\text{\tiny vac}}^{\text{\tiny
theory}}\approx-10^{97}\text{kg}/\text{m}^{3}$. Thus, it is possible that such
a large negative $\rho_{\text{\tiny vac}}^{\text{\tiny m=0}}\lll 0$
(associated with massless chiral matter or with gauge matter comprised of
paired massless fermions) can just nearly cancel a large positive
$\rho_{\text{\tiny vac}}^{\text{\tiny m>0}}\ggg 0$ (associated with massive
baryonic matter) leaving a small residual $\rho_{\text{\tiny
vac}}^{\text{\tiny obs}}\gtrsim 0$ that is experimentally observable. At ever
larger cuts, the value of the vacuum energy oscillates wildly about zero.
Nevertheless, the quantum lattice gas model appears to be one potential route
to reconcile a discretized quantum field theory, at least a version modified
at a small scale by (24), with the well accepted experimental observation of a
positive cosmological constant by employing a plausible wave number cutoff
parameter that corresponds to a fundamental grid scale. In a subsequent paper,
we will numerically evaluate the novel unitary collision operator (18b)
employed in a quantum algorithmic simulation of the dynamical behavior of a
system of Dirac particles.

##  VII Acknowledgements

Thanks to G. Vahala, J. Erlich, and N.H. Margolus for helpful comments.

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