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NT@UW-12-14

INT-PUB-12-046

Constraints on the Universe as a Numerical Simulation

Silas R. Beane,1, 2, ∗Zohreh Davoudi,3, †and Martin J. Savage3, ‡

1Institute for Nuclear Theory, Box 351550, Seattle, WA 98195-1550, USA

2Helmholtz-Institut für Strahlen- und Kernphysik (Theorie),

Universität Bonn, D-53115 Bonn, Germany

3Department of Physics, University of Washington,

Box 351560, Seattle, WA 98195, USA

(Dated: January 8, 2014 - 16:15)

Abstract

Observable consequences of the hypothesis that the observed universe is a numerical simulation

performed on a cubic space-time lattice or grid are explored. The simulation scenario is ﬁrst

motivated by extrapolating current trends in computational resource requirements for lattice QCD

into the future. Using the historical development of lattice gauge theory technology as a guide,

we assume that our universe is an early numerical simulation with unimproved Wilson fermion

discretization and investigate potentially-observable consequences. Among the observables that are

considered are the muon g−2and the current diﬀerences between determinations of α, but the

most stringent bound on the inverse lattice spacing of the universe, b−1>

∼1011 GeV, is derived

from the high-energy cut oﬀ of the cosmic ray spectrum. The numerical simulation scenario could

reveal itself in the distributions of the highest energy cosmic rays exhibiting a degree of rotational

symmetry breaking that reﬂects the structure of the underlying lattice.

∗beane@hiskp.uni-bonn.de. On leave from the University of New Hampshire.

†davoudi@uw.edu

‡savage@phys.washington.edu

1

arXiv:1210.1847v2 [hep-ph] 9 Nov 2012

I. INTRODUCTION

Extrapolations to the distant futurity of trends in the growth of high-performance com-

puting (HPC) have led philosophers to question —in a logically compelling way— whether

the universe that we currently inhabit is a numerical simulation performed by our distant

descendants [1]. With the current developments in HPC and in algorithms it is now pos-

sible to simulate Quantum Chromodynamics (QCD), the fundamental force in nature that

gives rise to the strong nuclear force among protons and neutrons, and to nuclei and their

interactions. These simulations are currently performed in femto-sized universes where the

space-time continuum is replaced by a lattice, whose spatial and temporal sizes are of the

order of several femto-meters or fermis (1 fm = 10−15 m), and whose lattice spacings (dis-

cretization or pixelation) are fractions of fermis 1. This endeavor, generically referred to as

lattice gauge theory, or more speciﬁcally lattice QCD, is currently leading to new insights

into the nature of matter 2. Within the next decade, with the anticipated deployment of

exascale computing resources, it is expected that the nuclear forces will be determined from

QCD, reﬁning and extending their current determinations from experiment, enabling pre-

dictions for processes in extreme environments, or of exotic forms of matter, not accessible

to laboratory experiments. Given the signiﬁcant resources invested in determining the quan-

tum ﬂuctuations of the fundamental ﬁelds which permeate our universe, and in calculating

nuclei from ﬁrst principles (for recent works, see Refs. [4–6]), it stands to reason that future

simulation eﬀorts will continue to extend to ever-smaller pixelations and ever-larger vol-

umes of space-time, from the femto-scale to the atomic scale, and ultimately to macroscopic

scales. If there are suﬃcient HPC resources available, then future scientists will likely make

the eﬀort to perform complete simulations of molecules, cells, humans and even beyond.

Therefore, there is a sense in which lattice QCD may be viewed as the nascent science of

universe simulation, and, as will be argued in the next paragraph, very basic extrapolation

of current lattice QCD resource trends into the future suggest that experimental searches

for evidence that our universe is, in fact, a simulation are both interesting and logical.

There is an extensive literature which explores various aspects of our universe as a simula-

tion, from philosophical discussions [1], to considerations of the limits of computation within

our own universe [7], to the inclusion of gravity and the standard model of particle physics

into a quantum computation [8], and to the notion of our universe as a cellular automa-

ton [9–12]. There have also been extensive connections made between fundamental aspects of

computation and physics, for example, the translation of the Church-Turing principle [13, 14]

into the language of physicists by Deutsch [15]. Finally, the observational consequences due

to limitations in accuracy or ﬂaws in a simulation have been considered [16]. In this work, we

take a pedestrian approach to the possibility that our universe is a simulation, by assuming

that a classical computer (i.e. the classical limit of a quantum computer) is used to simulate

the quantum universe (and its classical limit), as is done today on a very small scale, and ask

if there are any signatures of this scenario that might be experimentally detectable. Further,

we do not consider the implications of, and constraints upon, the underlying information,

and its movement, that are required to perform such extensive simulations. It is the case

that the method of simulation, the algorithms, and the hardware that are used in future

simulations are unknown, but it is conceivable that some of the ingredients used in present

1Surprisingly, while QCD and the electromagnetic force are currently being calculated on the lattice, the

diﬃculties in simulating the weak nuclear force and gravity on a lattice have so far proved insurmountable.

2See Refs. [2, 3] for recent reviews of the progress in using lattice gauge theory to calculate the properties

of matter.

2

SPECTRUM HcloverL

MILC HasqtadL

0

2

4

6

8

10

12

14

15

20

25

30

Years After 1999

Log@L5b6D

L=10-6m, b =0.1 fm

L=1 m, b =0.1 fm

0

20

40

60

80

100

120

140

0

50

100

150

200

250

Years After 1999

Log@L5b6D

FIG. 1. Left panel: linear ﬁt to the logarithm of the CRRs of MILC asqtad and the SPECTRUM

anisotropic lattice ensemble generations. Right panel: extrapolation of the ﬁt curves into the future,

as discussed in the text. The blue (red) horizontal line corresponds to lattice sizes of one micron

(meter), and vertical bands show the corresponding extrapolated years beyond 1999 for the lattice

generation programs.

day simulations of quantum ﬁelds remain in use, or are used in other universes, and so

we focus on one aspect only: the possibility that the simulations of the future employ an

underlying cubic lattice structure.

In contrast with Moore’s law, which is a statement about the exponential growth of

raw computing power in time, it is interesting to consider the historical growth of mea-

sures of the computational resource requirements (CRRs) of lattice QCD calculations, and

extrapolations of this trend to the future. In order to do so, we consider two lattice gener-

ation programs: the MILC asqtad program [17], which over a twelve year span generated

ensembles of lattice QCD gauge conﬁgurations, using the Kogut-Susskind [18] (staggered)

discretization of the quark ﬁelds, with lattice spacings, b, ranging from 0.18 to 0.045 fm, and

lattice sizes (spatial extents), L, ranging from 2.5to 5.8fm, and the on-going anisotropic

program carried out by the SPECTRUM collaboration [19], using the clover-Wilson [20, 21]

discretization of the quark ﬁelds, which has generated lattice ensembles at b∼0.1fm, with

Lranging from 2.4to 4.9fm [22]. At ﬁxed quark masses, the CRR of a lattice ensem-

ble generation (in units of petaFLOP-years) scales roughly as the dimensionless number

λQCD L5/b6, where λQC D ≡1fm is a typical QCD distance scale. In ﬁg. 1 (left panel), the

CRRs are presented on a logarithmic scale, where year one corresponds to 1999, when MILC

initiated its asqtad program of 2+1-ﬂavor ensemble generation. The bands are linear ﬁts

to the data. While the CRR curves in some sense track Moore’s law, they are more than

a statement about increasing FLOPS. Since lattice QCD simulations include the quantum

ﬂuctuations of the vacuum and the eﬀects of the strong nuclear force, the CRR curve is a

statement about simulating universes with realistic fundamental forces. The extrapolations

of the CRR trends into the future are shown in the right panel of ﬁg. 1. The blue (red)

horizontal line corresponds to a lattice of the size of a micro-meter (meter), a typical length

scale of a cell (human), and at a lattice spacing of 0.1fm. There are, of course, many caveats

to this extrapolation. Foremost among them is the assumption that an eﬀective Moore’s

Law will continue into the future, which requires technological and algorithmic developments

to continue as they have for the past 40 years. Related to this is the possible existence of

the technological singularity [23, 24], which could alter the curve in unpredictable ways.

3

And, of course, human extinction would terminate the exponential growth [1]. However,

barring such discontinuities in the curve, these estimates are likely to be conservative as

they correspond to full simulations with the fundamental forces of nature. With ﬁnite re-

sources at their disposal, our descendants will likely make use of eﬀective theory methods,

as is done today, to simulate every-increasing complexity, by, for instance, using meshes

that adapt to the relevant physical length scales, or by using ﬂuid dynamics to determine

the behavior of ﬂuids, which are constrained to rigorously reproduce the fundamental laws

of nature. Nevertheless, one should keep in mind that the CRR curve is based on lattice

QCD ensemble generation and therefore is indicative of the ability to simulate the quantum

ﬂuctuations associated with the fundamental forces of nature at a given lattice spacing and

size. The cost to perform the measurements that would have to be done in the background

of these ﬂuctuations in order to simulate —for instance— a cell could, in principle, lie on a

signiﬁcantly steeper curve.

We should comment on the simulation scenario in the context of ongoing attempts to

discover the theoretical structure that underlies the Standard Model of particle physics, and

the expectation of the uniﬁcation of the forces of nature at very short distances. There

has not been much interest in the notion of an underlying lattice structure of space-time

for several reasons. Primary among them is that in Minkowski space, a non-vanishing spa-

tial lattice spacing generically breaks space-time symmetries in such a way that there are

dimension-four Lorentz breaking operators in the Standard Model, requiring a large num-

ber of ﬁne-tunings to restore Lorentz invariance to experimentally veriﬁed levels [25]. The

fear is that even though Lorentz violating dimension four operators can be tuned away at

tree-level, radiative corrections will induce them back at the quantum level as is discussed

in Refs. [26, 27]. This is not an issue if one assumes the simulation scenario for the same

reason that it is not an issue when one performs a lattice QCD calculation 3. The under-

lying space-time symmetries respected by the lattice action will necessarily be preserved at

the quantum level. In addition, the notion of a simulated universe is sharply at odds with

the reductionist prejudice in particle physics which suggests the uniﬁcation of forces with a

simple and beautiful predictive mathematical description at very short distances. However,

the discovery of the string landscape [28, 29], and the current inability of string theory to

provide a useful predictive framework which would post-dict the fundamental parameters of

the Standard Model, provides the simulators (future string theorists?) with a purpose: to

systematically explore the landscape of vacua through numerical simulation. If it is indeed

the case that the fundamental equations of nature allow on the order of 10500 solutions [30],

then perhaps the most profound quest that can be undertaken by a sentient being is the ex-

ploration of the landscape through universe simulation. In some weak sense, this exploration

is already underway with current investigations of a class of conﬁning beyond-the-Standard-

Model (BSM) theories, where there is only minimal experimental guidance at present (for

one recent example, see Ref. [31]). Finally, one may be tempted to view lattice gauge the-

ory as a primitive numerical tool, and that the simulator should be expected to have more

eﬃcient ways of simulating reality. However, one should keep in mind that the only known

way to deﬁne QCD as a consistent quantum ﬁeld theory is in the context of lattice QCD,

which suggests a fundamental role for the lattice formulation of gauge theory.

Physicists, in contrast with philosophers, are interested in determining observable con-

sequences of the hypothesis that we are a simulation 4 5. In lattice QCD, space-time is

3Current lattice QCD simulations are performed in Euclidean space, where the underlying hyper-cubic

symmetry protects Lorentz invariance breaking in dimension four operators. However, Hamiltonian lattice

formulations, which are currently too costly to be practical, are also possible.

4There are a number of peculiar observations that could be attributed to our universe being a simulation,

4

replaced by a ﬁnite hyper-cubic grid of points over which the ﬁelds are deﬁned, and the

(now) ﬁnite-dimensional quantum mechanical path integral is evaluated. The grid breaks

Lorentz symmetry (and hence rotational symmetry), and its eﬀects have been deﬁned within

the context of a low-energy eﬀective ﬁeld theory (EFT), the Symanzik action, when the lat-

tice spacing is small compared with any physical length scales in the problem [33, 34] 6.

The lattice action can be modiﬁed to systematically improve calculations of observables, by

adding irrelevant operators with coeﬃcients that can be determined nonperturbatively. For

instance, the Wilson action can be O(b)-improved by including the Sheikholeslami-Wohlert

term [21]. Given this low-energy description, we would like to investigate the hypothesis

that we are a simulation with the assumption that the development of simulations of the

universe in some sense parallels the development of lattice QCD calculations. That is, early

simulations use the computationally “cheapest” discretizations with no improvement. In par-

ticular, we will assume that the simulation of our universe is done on a hyper-cubic grid 7

and, as a starting point, we will assume that the simulator is using an unimproved Wilson

action, that produces O(b)artifacts of the form of the Sheikholeslami-Wohlert operator in

the low-energy theory 8.

In section II, the simple scenario of an unimproved Wilson action is introduced. In

section III, by looking at the rotationally-invariant dimension-ﬁve operator arising from

this action, the bounds on the lattice spacing are extracted from the current experimental

determinations, and theoretical calculations, of g−2of the electron and muon, and from

the ﬁne-structure constant, α, determined by the Rydberg constant. Section IV considers

the simplest eﬀects of Lorentz symmetry breaking operators that ﬁrst appear at O(b2),

and modiﬁcations to the energy-momentum relation. Constraints on the energy-momentum

relation due to cosmic ray events are found to provide the most stringent bound on b. We

conclude in section V.

II. UNIMPROVED WILSON SIMULATION OF THE UNIVERSE

The simplest gauge invariant action of fermions which does not contain doublers is the

Wilson action,

S(W)=b4X

xL(W)(x) = b4m+4

bX

x

ψ(x)ψ(x)

+b3

2X

x

ψ(x)(γµ−1) Uµ(x)ψ(x+bˆµ)−(γµ+ 1) U†

µ(x−bˆµ)ψ(x−bˆµ),(1)

which describes a fermion, ψ, of mass minteracting with a gauge ﬁeld, Aµ(x), through the

gauge link,

Uµ(x) = exp ig ˆx+bˆµ

x

dzAµ(z),(2)

but that cannot be tested at present. For instance, it could be that the observed non-vanishing value of

the cosmological constant is simply a rounding error resulting from the number zero being entered into a

simulation program with insuﬃcient precision.

5Hsu and Zee [32] have suggested that the CMB provides an opportunity for a potential creator/simulator

of our universe to communicate with the created/simulated without further intervention in the evolution

of the universe. If, in fact, it is determined that observables in our universe are consistent with those that

would result from a numerical simulation, then the Hsu-Zee scenario becomes a more likely possibility.

Further, it would then become interesting to consider the possibility of communicating with the simulator,

5

where ˆµis a unit vector in the µ-direction, and gis the coupling constant of the theory.

Expanding the Lagrangian density, L(W), in the lattice spacing (that is small compared with

the physical length scales), and performing a ﬁeld redeﬁnition [38], it can be shown that the

Lagrangian density takes the form

L(W)=ψD/ψ + ˜mψψ +Cp

gb

4ψσµν Gµν ψ+O(b2),(3)

where Gµν =−i[Dµ, Dν]/g is the ﬁeld strength tensor and Dµis the covariant derivative. ˜m

is a redeﬁned mass which contains O(b)lattice spacing artifacts (that can be tuned away).

The coeﬃcient of the Pauli term ψσµν Gµν ψis ﬁxed at tree level, Cp= 1 + O(α), where

α=g2/(4π). It is worth noting that as is usual in lattice QCD calculations, the lattice

action can be O(b)improved by adding a term of the form δL(W)=Csw gb

4ψσµν Gµν ψto the

Lagrangian with Csw =−Cp+O(α). This is the so-called Sheikholeslami-Wohlert term.

Of course there is no reason to assume that the simulator had to have performed such an

improvement in simulating the universe.

III. ROTATIONALLY INVARIANT MODIFICATIONS

Lorentz symmetry is recovered in lattice calculations as the lattice spacing vanishes when

compared with the scales of the system. It is useful to consider contributions to observables

from a non-zero lattice spacing that are Lorentz invariant and consequently rotationally

invariant, and those that are not. While the former type of modiﬁcations could arise from

many diﬀerent BSM scenarios, the latter, particularly modiﬁcations that exhibit cubic sym-

metry, would be suggestive of a structure consistent with an underlying discretization of

space-time.

1. QED Fine Structure Constant and the Anomalous Magnetic Moment

For our present purposes, we will assume that Quantum Electrodynamics (QED) is simulated

with this unimproved action, eq. (1). The O(b)contribution to the lattice action induces an

additional contribution to the fermion magnetic moments. Speciﬁcally, the Lagrange density

that describes electromagnetic interactions is given by eq. (3), where the interaction with

an external magnetic ﬁeld Bis described through the covariant derivative Dµ=∂µ+ie ˆ

QAµ

with e > 0and the electromagnetic charge operator ˆ

Q, and where the vector potential

satisﬁes ∇ × A=B. The interaction Hamiltonian density in Minkowski-space is given by

Hint =e

2mψAµ(i−→

∂µ−i←−

∂µ)ˆ

Qψ +ˆ

Qe

4mψσµν Fµν ψ+Cp

ˆ

Qeb

4ψσµν Fµν ψ+... . (4)

where Fµν =∂µAν−∂νAµis the electromagnetic ﬁeld strength tensor, and the ellipses denote

terms suppressed by additional powers of b. By performing a non-relativistic reduction, the

or even more interestingly, manipulating or controlling the simulation itself.

6The ﬁnite volume of the hyper-cubic grid also breaks Lorentz symmetry. A recent analysis of the CMB

suggests that universe has a compact topology, consistent with two compactiﬁed spatial dimensions and

with a greater than 4σdeviation from three uncompactiﬁed spatial dimensions [35].

7The concept of the universe consisting of ﬁelds deﬁned on nodes, and interactions propagating along the

links between the nodes, separated by distances of order the Planck length, has been considered previously,

e.g. see Ref. [36].

8It has been recently pointed out that the domain-wall formulation of lattice fermions provides a mecha-

6

last two terms in eq. 4 give rise to Hint,mag =−µ·B, where the electron magnetic moment

µis given by

µ=ˆ

Qe

2m(g+ 2mb Cp+...)S=g(b)ˆ

Qe

2mS,(5)

where gis the usual fermion g-factor and Sis its spin. Note that the lattice spacing

contribution to the magnetic moment is enhanced relative to the Dirac contribution by one

power of the particle mass.

For the electron, the eﬀective g-factor has an expansion at ﬁnite lattice spacing of

g(e)(b)

2= 1 + C2α

π+C4α

π2+C6α

π3+C8α

π4+C10 α

π5

+ahadrons +aµ,τ +aweak +mebCp+... , (6)

where the coeﬃcients Ci, in general, depend upon the ratio of lepton masses. The calculation

by Schwinger provides the leading coeﬃcient of C2=1

2. The experimental value of g(e)

expt/2 =

1.001 159 652 180 73(28) gives rise to the best determination of the ﬁne structure constant

α(at b= 0) [39]. However, when the lattice spacing is non-zero, the extracted value of α

becomes a function of b,

α(b) = α(0) −2πmebCp+Oα2b,(7)

where α(0)−1= 137.035 999 084(51) is determined from the experimental value of electron

g-factor as quoted above. With one experimental constraint and two parameters to deter-

mine, αand b, unique values for these quantities cannot be established, and an orthogonal

constraint is required. One can look at the muon g−2which has a similar QED expansion

to that of the electron, including the contribution from the non-zero lattice spacing,

g(µ)(b)

2= 1 + C(µ)

2α

π+C(µ)

4α

π2+C(µ)

6α

π3+C(µ)

8α

π4+C(µ)

10 α

π5

+a(µ)

hadrons +a(µ)

e,τ +a(µ)

weak +mµbCp+... . (8)

Inserting the electron g−2(at ﬁnite lattice spacing) gives

g(µ)(b)

2=g(µ)(0)

2+ (mµ−me)bCp+Oα2b.(9)

Given that the standard model calculation of g(µ)(0) is consistent with the experimental

value, with a ∼3.6σdeviation, one can put a limit on bfrom the diﬀerence and uncer-

tainty in theoretical and experimental values of g(µ),g(µ)

expt/2 = 1.001 165 920 89(54)(33) and

g(µ)

theory/2 = 1.001 165 918 02(2)(42)(26) [39]. Attributing this diﬀerence to a ﬁnite lattice

spacing, these values give rise to

b−1= (3.6±1.1) ×107GeV ,(10)

which provides an approximate upper bound on the lattice spacing.

nism by which the number of generations of fundamental particles is tied to the form of the dispersion

relation [37]. Space-time would then be a topological insulator.

7

2. The Rydberg Constant and α

Another limit can be placed on the lattice spacing from diﬀerences between the value of α

extracted from the electron g−2and from the Rydberg constant, R∞. The latter extraction,

as discussed in Ref. [39], is rather complicated, with the value of the R∞obtained from a χ2-

minimization ﬁt involving the experimentally determined energy-level splittings. However, to

recover the constraints on the Dirac energy-eigenvalues (which then lead to R∞), theoretical

corrections must be ﬁrst removed from the experimental values. To begin with, one can

obtain an estimate for the limit on bby considering the diﬀerences between α’s obtained

from various methods assuming that the only contributions are from QED and the lattice

spacing. Given that it is the reduced mass (µ∼me) that will compensate the lattice spacing

in these QED determinations (for an atom at rest in the lattice frame), one can write

δα = 2πmeb˜

Cp,(11)

where ˜

Cpis a number O(1) by naive dimensional analysis, and is a combination of the con-

tributions from the two independent extractions of α. There is no reason to expect complete

cancellation between the contributions from two diﬀerent extractions. In fact, it is straight-

forward to show that the O(b)contribution to the value of αdetermined from the Rydberg

constant is suppressed by α4m2

e, and therefore the above assumption is robust. In addition to

the electron g−2determination of ﬁne structure constant as quoted above, the next precise

determination of αcomes form the atomic recoil experiments, α−1= 137.035 999 049(90) 9

[39], given an a priori determined value of the Rydberg constant. This gives rise to a diﬀer-

ence of |δα|= (1.86 ±5.51) ×10−12 between two extractions, which translates into

b=|(−0.6±1.7) ×10−9|GeV−1.(12)

As this result is consistent with zero, the 1σvalues of the lattice spacing give rise to a limit

of

b−1>

∼4×108GeV ,(13)

which is seen to be an order of magnitude more precise than that arising from the muon

g−2.

For more sophisticated simulations in which chiral symmetry is preserved by the lattice

discretization, the coeﬃcient Cpwill vanish or will be exponentially small. As a result,

the bound on the lattice spacing derived from the muon g−2and from the diﬀerences

between determinations of αwill be signiﬁcantly weaker. In these analyses, we have worked

with QED only, and have not included the full electroweak interactions as chiral gauge

theories have not yet been successfully latticized. Consequently, these constraints are to be

considered estimates only, and a more complete analysis needs to be performed when chiral

gauge theories can be simulated.

IV. ROTATIONAL SYMMETRY BREAKING

While there are more conventional scenarios for BSM physics that generate deviations in

g−2from the standard model prediction, or diﬀerences between independent determinations

9Extracted from a 87Rb recoil experiment [40].

8

of α, the breaking of rotational symmetry would be a solid indicator of an underlying space-

time grid, although not the only one. As has been extensively discussed in the literature,

another scenario that gives rise to rotational invariance violation involves the introduction of

an external background with a preferred direction. Such a preferred direction can be deﬁned

via a ﬁxed vector, uµ[41]. The eﬀective low-energy Lagrangian of such a theory contains

Lorentz covariant higher dimension operators with a coupling to this background vector, and

breaks both parity and Lorentz invariance [42]. Dimension three, four and ﬁve operators,

however, are shown to be severely constrained by experiment, and such contributions in the

low-energy action (up to dimension ﬁve) have been ruled out [25, 41, 43, 44].

3. Atomic Level Splittings

At O(b2)in the lattice spacing expansion of the Wilson action, that is relevant to describing

low-energy processes, there is a rotational-symmetry breaking operator that is consistent

with the lattice hyper-cubic symmetry,

LRV =CRV b2

6

4

X

µ=1

ψ γµDµDµDµψ , (14)

where the tree-level value of CRV = 1. In taking matrix elements of this operator in the

Hydrogen atom, where the binding energy is suppressed by a factor of αcompared with

the typical momentum, the dominant contribution is from the spatial components. As each

spatial momentum scales as meα, in the non-relativistic limit, shifts in the energy levels are

expected to be of order

δE ∼CRV α4m3

eb2.(15)

To understand the size of energy splittings, a lattice spacing of b−1= 108GeV gives an

energy shift of order δE ∼10−26 eV, including for the splittings between substates in

given irreducible representations of SO(3) with angular momentum J≥2. This magnitude

of energy shifts and splittings is presently unobservable. Given present technology, and

constraints imposed on the lattice spacing by other observables, we conclude that there is

little chance to see such an eﬀect in the atomic spectrum.

4. The Energy-Momentum Relation and Cosmic Rays

Constraints on Lorentz-violating perturbations to the standard model of electroweak interac-

tions from diﬀerences in the maximal attainable velocity (MAV) of particles (e.g. Ref. [25]),

and on interactions with a non-zero vector ﬁeld (e.g. Ref. [45]), have been determined pre-

viously. Assuming that each particle satisﬁes an energy-momentum relation of the form

E2

i=|pi|2c2

i+m2

ic4

i(along with the conservation of both energy and momentum in any

given process), if cγexceeds ce±, the process γ→e+e−becomes possible for photons with

an energy greater than the critical energy Ecrit.= 2mec2

ecγ/qc2

γ−c2

e±, and the observa-

tion of high energy primary cosmic photons with Eγ<

∼20 TeV translates into the constraint

cγ−ce±<

∼10−15. Ref. [25] presents a series of exceedingly tight constraints on diﬀerences be-

tween the speed of light between diﬀerent particles, with typical sizes of δcij <

∼10−21 −10−22

9

FIG. 2. The energy surface of a massless, non-interacting Wilson fermion with r= 1 as a function

of momentum in the x and y directions, bounded by −π < bpx,y < π, for pz= 0 is shown in blue.

The continuum dispersion relation is shown as the red surface.

for particles of species iand j. At ﬁrst glance, these constraints [26] would appear to

also provide tight constraints on the size of the lattice spacing used in a simulation of the

universe. However, this is not the case. As the speed of light for each particle in the dis-

cretized space-time depends on its three-momentum, the constraints obtained by Coleman

and Glashow [25] do not directly apply to processes occurring in a lattice simulation.

The dispersion relations satisﬁed by bosons and Wilson fermions in a lattice simulation

(in Minkowski space) are

sinh2(bEb

2)−X

j=1,2,3

sin2(bkj

2)−(bmb

2)2= 0 ;

Eb=q|k|2+m2

b+O(b2),(16)

and

sinh2(bEf)−X

j=1,2,3

sin2(bkj)−"bmf+ 2r X

j=1,2,3

sin2(bkj

2)−sinh2(bEf

2)!#2

= 0 ;

Ef=q|k|2+m2

f−r b m3

f

2q|k|2+m2

f

+O(b2),(17)

respectively, where ris the coeﬃcient of the Wilson term, Eband Efare the energy of a

boson and fermion with momentum k, respectively. The summations are performed over

the components along the lattice Cartesian axes corresponding to the x,y, and z spatial

directions. The implications of these dispersion relations for neutrino oscillations along one

of the lattice axes have been considered in Ref. [46]. Further, they have been considered as

a possible explanation [47] of the (now retracted) OPERA result suggesting superluminal

neutrinos [48]. The violation of Lorentz invariance resulting from these dispersion relations

is due to the fact that they have only cubic symmetry and not full rotational symmetry, as

shown in ﬁg. 2. It is in the limit of small momentum, compared to the inverse lattice spacing,

that the dispersion relations exhibit rotational invariance. While for the fundamental parti-

cles, the dispersion relations in eq. (16) and eq. (17) are valid, for composite particles, such

as the proton or pion, the dispersion relations will be dynamically generated. In the present

analysis we assume that the dispersion relations for all particles take the form of those in

10

eq. (16) and eq. (17). It is also interesting to note that the polarizations of the massless

vector ﬁelds are not exactly perpendicular to their direction of propagation for some direc-

tions of propagation with respect to the lattice axes, with longitudinal components present

for non-zero lattice spacings.

Consider the process p→p+γ, which is forbidden in the vacuum by energy-momentum

conservation in special relativity when the speed of light of the proton and photon are equal,

cp=cγ. Such a process can proceed in-medium when vp> cγ, corresponding to Cerenkov

radiation. In the situation where the proton and photon have diﬀerent MAV’s, the absence

of this process in vacuum requires that |cp−cγ|<

∼10−23 [25, 49]. In lattice simulations of

the universe, this process could proceed in the vacuum if there are ﬁnal state momenta

which satisfy energy conservation for an initial state proton with energy Eimoving in some

direction with respect to the underlying cubic lattice. Numerically, we ﬁnd that there are

no ﬁnal states that satisfy this condition, and therefore this process is forbidden for all

proton momentum 10. In contrast, the process γ→e+e−, which provides tight constraints

on diﬀerences between MAV’s [25], can proceed for very high energy photons (those with

energies comparable to the inverse lattice spacing) near the edges of the Brillouin zone.

Further, very high energy π0’s are stable against π0→γγ, as is the related process γ→π0γ.

With the dispersion relation of special relativity, the structure of the cosmic ray spec-

trum is greatly impacted by the inelastic collisions of nucleons with the cosmic microwave

background (CMB) [50, 51]. Processes such as γCMB +N→∆give rise to the predicted

GKZ-cut oﬀ scale [50, 51] of ∼6×1020 eV in the spectrum of high energy cosmic rays. Recent

experimental observations show a decline in the ﬂuxes starting around this value [52, 53],

indicating that the GKZ-cut oﬀ (or some other cut oﬀ mechanism) is present in the cosmic

ray ﬂux. For lattice spacings corresponding to an energy scale comparable to the GKZ cut

oﬀ, the cosmic ray spectrum will exhibit signiﬁcant deviations from isotropy, revealing the

cubic structure of the lattice. However, for lattice spacings much smaller than the GKZ

cut oﬀ scale, the GKZ mechanism cuts oﬀ the spectrum, eﬀectively hiding the underlying

lattice structure. When the lattice rest frame coincides with the CMB rest frame, head-on

interactions between a high energy proton with momentum |p|and a photon of (very-low)

energy ωcan proceed through the ∆resonance when

ω=m2

∆−m2

N

4|p|"1 + √πb2|p|2

9 Y0

4(θ, φ) + r5

14 Y+4

4(θ, φ) + Y−4

4(θ, φ)!#

−m3

∆−m3

N

4|p|br +... , (18)

for |p| 1/b, where θand φare the polar and azimuthal angles of the particle momenta in

the rest frame of the lattice, respectively. This represents a lower bound for the energy of

photons participating in such a process with arbitrary collision angles.

The lattice spacing itself introduces a cut oﬀ to the cosmic ray spectrum. For both the

fermions and the bosons, the cut oﬀ from the dispersion relation is Emax ∼1/b. Equating

this to the GKZ cut oﬀ corresponds to a lattice spacing of b∼10−12 fm, or a mass scale of

b−1∼1011 GeV. Therefore, the lattice spacing used in the lattice simulation of the universe

10 A more complete treatment of this process involves using the parton distributions of the proton to relate

its energy to its momentum [26]. For the composite proton, the p→p+γprocess becomes kinematically

allowed, but with a rate that is suppressed by O(Λ8

QCDb7)due to the momentum transfer involved,

eﬀectively preventing the process from occuring. With momentum transfers of the scale ∼1/b, the ﬁnal

states that would be preferred in inclusive decays, p→Xh+γ, are kinematically forbidden, with invariant

masses of ∼1/b. More reﬁned explorations of this and other processes are required.

11

must be b<

∼10−12 fm in order for the GZK cut oﬀ to be present or for the lattice spacing

itself to provide the cut oﬀ in the cosmic ray spectrum. The most striking feature of the

scenario in which the lattice provides the cut oﬀ to the cosmic ray spectrum is that the

angular distribution of the highest energy components would exhibit cubic symmetry in the

rest frame of the lattice, deviating signiﬁcantly from isotropy. For smaller lattice spacings,

the cubic distribution would be less signiﬁcant, and the GKZ mechanism would increasingly

dominate the high energy structure. It may be the case that more advanced simulations will

be performed with non-cubic lattices. The results obtained for cubic lattices indicate that

the symmetries of the non-cubic lattices should be imprinted, at some level, on the high

energy cosmic ray spectrum.

V. CONCLUSIONS

In this work, we have taken seriously the possibility that our universe is a numerical simula-

tion. In particular, we have explored a number of observables that may reveal the underlying

structure of a simulation performed with a rigid hyper-cubic space-time grid. This is mo-

tivated by the progress in performing lattice QCD calculations involving the fundamental

ﬁelds and interactions of nature in femto-sized volumes of space-time, and by the simulation

hypothesis of Bostrom [1]. A number of elements required for a simulation of our universe

directly from the fundamental laws of physics have not yet been established, and we have

assumed that they will, in fact, be developed at some point in the future; two important

elements being an algorithm for simulating chiral gauge theories, and quantum gravity. It

is interesting to note that in the simulation scenario, the fundamental energy scale deﬁned

by the lattice spacing can be orders of magnitude smaller than the Planck scale, in which

case the conﬂict between quantum mechanics and gravity should be absent.

The spectrum of the highest energy cosmic rays provides the most stringent constraint

that we have found on the lattice spacing of a universe simulation, but precision measure-

ments, particularly the muon g−2, are within a few orders of magnitude of being sensitive

to the chiral symmetry breaking aspects of a simulation employing the unimproved Wilson

lattice action. Given the ease with which current lattice QCD simulations incorporate im-

provement or employ discretizations that preserve chiral symmetry, it seems unlikely that

any but the very earliest universe simulations would be unimproved with respect to the

lattice spacing. Of course, improvement in this context masks much of our ability to probe

the possibility that our universe is a simulation, and we have seen that, with the excep-

tion of the modiﬁcations to the dispersion relation and the associated maximum values of

energy and momentum, even O(b2)operators in the Symanzik action easily avoid obvious

experimental probes. Nevertheless, assuming that the universe is ﬁnite and therefore the

resources of potential simulators are ﬁnite, then a volume containing a simulation will be

ﬁnite and a lattice spacing must be non-zero, and therefore in principle there always remains

the possibility for the simulated to discover the simulators.

Acknowledgments

We would like to thank Eric Adelberger, Blayne Heckel, David Kaplan, Kostas Orginos,

Sanjay Reddy and Kenneth Roche for interesting discussions. We also thank William Det-

mold, Thomas Luu and Ann Nelson for comments on earlier versions of the manuscript. SRB

12

was partially supported by the INT during the program INT-12-2b: Lattice QCD studies of

excited resonances and multi-hadron systems, and by NSF continuing grant PHY1206498.

In addition, SRB gratefully acknowledges the hospitality of HISKP and the support of the

Mercator programme of the Deutsche Forschungsgemeinschaft. ZD and MJS were supported

in part by the DOE grant DE-FG03-97ER4014.

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