Title Regarding “the Weyl Tile Argument”

Synopsis Space (or spacetime) is called discrete just in case it is composed of

extended indivisible regions—call them “tiles.” Weyl (1949) famously argued

that if space were discrete, then Euclidean geometry cannot hold even approxi-

mately, which contradicts our observations. Since then, many philosophers have

proposed solutions to this argument (e.g., Bendegem 1987, Forrest 1995). While

these authors have successfully suggested some alternative accounts of geometry—

attractive or not—that allow for discrete space, they have missed an importantly

flawed assumption in Weyl’s argument: physical geometry is determined by fun-

damental spacetime structures independently from the dynamic laws. In this pa-

per, I aim to show its falsity through two rigorous examples: random walk in

statistic physics and quantum mechanics. These are intended to be proof of con-

cept for the claim that physical geometry arises from dynamic laws that do not

assume any metric notion (“metric” is a more general and technical term for dis-

tances). Even if space (or spacetime) is discrete, depending on dynamic laws, our

observations will be approximately the same as in Euclidean geometry. This is why

Weyl’s argument fails.

In the case of random walk, I show that on a two dimensional discrete space

represented by Z2 (pairs of integers), the probability distribution of a wander-

ing cat starting at any given position showing up at each tile is approximately

rotational invariant (it’s called random walk because, from any tile, the cat ran-

domly walks to one of its four neighboring tiles). If such probabilities are the only

observable quantities, then we have an embedding map from the discrete space

to Euclidean space that approximately preserves all structures and observations.

This means that Euclidean geometry is approximately recovered on the empirical

level.

1

In the case of quantum mechanics, I show that for a quantum mechanical sys-

tem starting with a sufficiently spread-out position wavefunction, the amplitude

of it reaching each tile at a later time is approximately rotational invariant. Assum-

ing that the probability of a quantum mechanical system in a region at a time is

the only kind of observational quantity, this again means that Euclidean geometry

is approximately recovered on the observational level. The case of random walk

is chosen for its rigor and conceptual clarity while the case of quantum mechanics

adds more physical relevance.

Based on these cases, I elaborate on and criticize Weyl’s two “geometricist” as-

sumptions (geometricism is roughly the view that geometry is more fundamental

than dynamics; the opposite view is called dynamicism): (1) large-scale or observ-

able physical geometry is determined by fundamental spacetime structures in-

dependent of dynamic laws; (2) some geometric structures (including the metric

structure) are ontologically and explanatorily prior to dynamics and must be pre-

sumed by the latter. The two cases show that, against (1), dynamic laws play a

fundamental role in determining observational geometry. Against (2), no metric

notion is presumed by the dynamic laws. It is also important to note that, while

the two cases are highly simplified toy examples, they are not contrived, ad-hoc

or overly complicated. They indicate that it is entirely possible that our actual

fundamental laws can similarly give rise to Euclidean geometry on the ordinary

scales. Thus, we should reject the geometricist assumptions in our understanding

of spacetime. While this paper largely focuses on criticizing Weyl’s argument, this

general lesson about physical geometry is what I aim at.

2

Regarding “the Weyl Tile Argument”

Draft

Prepared for APA pacific colloquium

Abstract Weyl (1949) famously argued that if space (or spacetime) were dis-

crete, then Euclidean geometry cannot hold even approximately. Since then, many

philosophers have proposed solutions to this argument (e.g., van Bendegem 1987,

Forrest 1995). While these authors have successfully suggested some alternative

accounts of geometry—attractive or not—that allow for discrete space, they have

missed an importantly flawed assumption in Weyl’s argument: physical geometry

is determined by fundamental spacetime structures independently from the dy-

namic laws. In this paper, I aim to show its falsity through two rigorous examples:

random walk in statistic physics and quantum mechanics.

Word Counts 3000 words (excluding the appendices)

Space is discrete just in case it is composed of indivisible extended regions. The

Weyl tile argument is an influential argument against discrete space. It purports to

show that if space were discrete, then Euclidean geometry would not be approxi-

mately true even at the ordinary scale. Why is that? Weyl argued: “If a square is

built up of miniature tiles, then there are as many tiles along the diagonal as there

are along the side; thus the diagonal should be equal in length to the side.” (Weyl

1949: 43) But if the Pythagorean theorem is approximately true, then the diagonal

3

should be about√

2 times as long as the side. It follows that Euclidean geometry

cannot emerge from the tile space at any scale (since the argument applies to any

number of tiles). But our space is approximately Euclidean at least at the ordinary

scale. So Weyl concluded that it could not be discrete.

This argument relies on the implicit assumption that the distance between any

two tiles is equal to the number of tiles between them. This is a natural assumption

endorsed by many authors, including Riemann (1866). This assumption is explic-

itly discussed and rejected in Chen (2020) as well as in van Bendegem (1987, 1995).

Other authors such as Forrest (1995) maintain this assumption but propose more

complicated models. (Note that the Weyl tile argument cannot be solved by simply

replacing the square tile arrangement with other simple regular arrangements—

Fritz (2012) shows that in order to approximate Euclidean geometry at the large

scale, the arrangement of the tiles need to be very irregular and complicated.)

These solutions have their merits and weaknesses.1 But they all fail to point out a

serious problem with Weyl’s assumption.

In this paper, I will argue that Weyl’s assumption about how large scale dis-

tances emerge from fundamental structures is wrong. It is wrong because it fails

to consider the imminently relevant possibility that large-scale distances emerge

from dynamic systems rather than the fundamental metric structure that exists

independent of the dynamics. To illustrate, I will give two concrete examples (ran-

dom walk in Section 1, and quantum mechanics in Section 2) as proofs of concept

according to which we can have apparent large-scale geometry without assuming

1As it will become apparent, this paper is not about criticizing these solutions. (My aim is tooffer a different, more critical response to Weyl’s argument that will provide additional insights.)But if it is of any interest, I can very briefly talk about what I perceive as some of the weaknesses ofthese solutions and why I feel unsatisfied. Forrest (1995) is discussed and criticized in Chen (2020),which I agree with, but the latter also faces the problem of an overly complicated and contrivedaccount of discrete space (in her account, real-valued primitive distances are held by a vast numberof “neighboring” tiles). van Bendegem (1995) discussed and criticized his own (1987) proposal, buthis new proposal has a complicated ideology: he posited both a set of microscopic geometry entities(“t-point,” “t-line”, etc.) and macroscopic ones (“point,” “line,” etc.), and the latter are governedby principles not reducible to those of the former.

4

any notion of distances at the fundamental level—and even if we assume Weyl’s

tile geometry at the fundamental level, it would not have empirical consequences.

The examples show that, depending on the dynamic laws, in discrete space, dy-

namic systems can still behave approximately the same as in Euclidean space un-

der certain conditions. Therefore, Weyl’s argument is fundamentally flawed.

But the point of the paper is not just to argue against Weyl’s argument and de-

fend discrete space, but also to support the view that geometry is not necessarily

prior to (or presumed by) dynamics in our physical reality. The position that ge-

ometry is not prior to but derived from dynamic laws is called dynamicism. The

opposite view that geometry comes first is called geometricism, which has a long

historical tradition and is still considered standard. For example, even though

Newton advocated the epistemological primacy of dynamics over geometry, he

still conceded to the ontological primacy of geometry (Principia, xvii-xviii).2 Ear-

man (1989) also famously said that “the laws of motion cannot be written on thin

air alone, but require the support of various spacetime structures” (46). There is

a heated debate in the last few decades between geometricism and dynamicism

especially in the context of relativistic theories (for example, see Brown 2005, Nor-

ton 2008; also see [Anonymized] for more details). Part of my purpose is to point

out the mutually beneficial connection between discrete space (or spacetime) and

dynamicism. (But I do not expect the readers of this paper to have any prior con-

sideration of this debate or of dynamicism.)

2We should distinguish this position (i.e., geometricism) from substantivalism, which Newtonallegedly subscribed to. There are two things worth clarifying here. First, substantivalism doesnot necessarily entail geometricism, nor does geometricism entail substantivalism. For the former,this paper provides an example: there is discrete space but it does not have a fundamental metricstructure. For the latter, (classical) relationalism is an example, according to which space does notexist, but material bodies have fundamental geometric relations. Second, Newton’s position shouldbe distinguished from standard substantivalism according to which spacetime is a substance andspacetime points are individuals. For example, see Newton “De Gravitatione”in Hall and Hall(1962); see also Disalle (1994).

5

1 RandomWalk

In this section, I will present the toy example of random walk to illustrate how

Euclidean geometry can emerge from dynamics that does not presume it. Random

walks are studied in statistical physics, which is a branch of physics that studies

how properties of macroscopic systems arise from stochastic microscopic motions.

As a proof of concept, this example is aimed at conceptual clarity and mathe-

matical rigor, though not depicting a physically realistic situation. I will discuss

its implications after presenting the technical result (the mathematical details are

left to Appendix A).

Following Weyl, I will focus on the two-dimensional square tile space in this

example. Let every tile be represented by a member of Z2. Imagine that there is

a tiny cat running from tile to tile. From each tile, the cat can move along four

directions, characterized by two basis vectors ±e1,±e2 (with |ei |=1 for all i).3 To

capture this, we can also postulate a topology on the tile space defined through

the primitive notion of “connectedness” which is irreflexive and symmetric (see

Roeper 1995 for a topological framework based on “connectness”):

Connectedness. For any tiles x,y ∈ Z2, they are connected iff |x − y| = 1.

Intuitively, every tile is connected to its neighboring four tiles (i.e., its left, right,

up, and down tiles). There is no metric structure on top of this. For convenience, I

will define the Euclidean distance between two tiles x,y ∈ Z2 to be |x − y| (note that

this is just a definition, not a postulation of a geometric structure). We can prove

the following:4

Isotropy. For any starting position x, and for any two tiles that have

approximately the same Euclidean distance to x, the probabilities of the

3For any x = (z1, z2) ∈ Zn, |x| =√z2

1 + z22.

4The proof is initially given to me by [anonymized] in correspondence, which I have checked tobe valid. Without his contribution, the current paper would be impossible.

6

cat showing up in them are approximately the same after sufficiently

long time (namely after the cat has taken sufficiently many steps).5

Let the starting position of the cat be the origin (0,0). It’s easy to see that, after

the first step, the probability of the cat showing up in each of the four neighboring

tiles is 1/4. Our goal is to calculate the probability for any tile after sufficiently

many steps. If the cat could not backtrack and only move in two directions, then

such a probability would be easy to calculate. In that case, to reach (x,y) after n

steps, the cat needs to take a total of x steps in the “horizontal” direction and a

total of y steps in the “vertical” direction. Using basic combinatorics, the result

would be (1/2)n(nx

), since each path the cat might take has a probability of (1/2)n

and there are(nx

)possible paths. The problem is trickier when the cat can move

back and forth along horizontal and vertical directions.6 Fortunately, with a neat

trick, we can obtain that the probability of reaching (x,y) after n steps is equal to

(1/4)n(

n(n+ x+ y)/2

)(n

(n+ x − y)/2

)(1)

See Appendix A.1 for the proof.

What is the result (1) equal to numerically? Gallager (1968) showed that the

binomial(nk

)has an upper bound

√n

2πk(n−k)enh(k/n) (k ∈ [1,n − 1] ∩ Z), where h =

−x lnx − (1 − x) ln(1 − x), and the binomial converges to the upper bound when n

5This result is similar one of the central theorems in probability theory, the central limit theorem(CLT), according to which the normalized sum of independent random variables (under certainconditions) tends towards a normal distribution (see van der Vaart 1998). We can apply this theo-rem to our case of random walk. CLT implies that as time tends to infinity, the distribution of theprobability converges to that of the n-dimensional normal distribution centered around the start-ing position. More specifically, the probability of the cat showing up at region around a particulartile (with the size O(

√n) where n is the number of the steps) is rotational invariant as time tends to

infinity. Isotropy is stronger than this general result because it is about the approximate isotropyof the probability distribution on single tiles rather than large regions.

6The obvious idea in the simple case would not work. For example, let the number of stepsalong the four directions be a,b,c,d respectively. The probability of reaching (x,y) after n steps isequal to (1/4)n

(na

)(n−ab

)(n−a−bc

), with a+ b = x and c+ d = y. But this probability cannot be calculated

because there are too many unknown variables.

7

and n− k are both very large. To highlight:

(nk

)→

√n

2πk(n− k)enh(k/n) (2)

Plugging the result (1) into this formula (2), with steps laid out in Appendix A.2,

we can obtain the following result.7 Assuming x � n and y � n, (1) is approxi-

mately equal to

2e(x2+y2)/n

π√n2 − (x2 + y2)

(3)

Notice that the result (3) only contains the Euclidean form x2 + y2 and is there-

fore rotational invariant. This can be straightforwardly generalized to any tile as

the starting position (as it is essentially a matter of coordinate translation). This

concludes the proof for Isotropy.

It follows from Isotropy that there is an embedding from the tile space into Eu-

clidean space that approximately preserves the probability distribution of the cat

after sufficiently long time. Assuming that such probabilities are our only obser-

vations, this means that our observations would be about the same as in Euclidean

space.8 In this sense, the apparent Euclidean geometry emerges from the tile space

under the dynamics of random walks.

To take stock, I would like to highlight some important features of this exam-

ple. First, in this example, there is no isotropic (or approximately isotropic) space-

time structure existing prior to dynamics. Indeed, I haven’t defined a metric struc-

ture, and the topology is far from isotropic. It is worth emphasizing that the notion

7The result is also double-checked numerically in computer programs.8Isotropy is stronger than what we need for recovering approximately Euclidean large scale

observations because more realistically, we can only observe the probability of the cat in a largeregion rather than a single tile. This theorem ensures that such probabilities will also be distributedin an approximately Euclidean way, but not vice versa. That is, even if the probability distributionsover tiles is not approximately Euclidean, the probability distributions averaging over large regionsmay still be approximately Euclidean.

8

of “Euclidean distance” is just a feature of how we represent the tile space, which

allows us to conveniently specify the embedding just mentioned, rather than re-

ferring to any primitive geometric structure. Importantly, it is not involved in

the dynamic laws. Second, the dynamic laws involved are not contrived or overly

complicated. Indeed, they are the simplest ones studied in statistical field theory.

These two features make this case interestingly different from the general case of

(say) simulating continuous mechanics on discrete pixels in computer programs.

The discrete pixels are embedded in the Euclidean space, and the programming

makes use of Euclidean geometry—or else the rules would be very complicated.

Furthermore, notice that the recovery of isotropy in this case does not critically

involve macroscopic regions consisting of many tiles. Rather, the statistical corre-

lations between individual tiles are already isotropic. This makes the case radically

different from—for example—Bendegem’s (1987, 1995) proposals, which crucially

rely on macroscopic geometric entities for recovering Euclidean geometry.

Of course, the case of random walk is physically unrealistic: it does not depict

the behavior of any fundamental particle or field; our space (or spacetime) is not

two dimensional; actual observations are vastly more complicated; etc. The most

relevant physical situations to this case are those studied by statistical field theory,

such as the transmission of heat (the transmission of heat would be isotropic even

if the molecules involves moved along discrete tiles). But this is not a realistic

interpretation, since the molecular movements occur at much larger scales than

the fundamental unit of space.

Nonetheless, this case already constitutes serious evidence against the implicit

assumption in the Weyl tile argument that the empirically observable distances are

determined by counting the number of tiles. Call dynamic laws isotropy-generating

just in case they satisfy some suitable version of Isotropy—that is, just in case the

dynamic laws defined on simple discrete space (with no metrics) nevertheless give

9

rise to rotational invariant observations. The case of random walk shows that it is a

very natural possibility that our fundamental laws are isotropy generating. Thus,

the fact that Weyl’s argument misses this possibility is a serious problem.

While I intend this case to do the conceptual heavy-lifting for my objection to

the Weyl tile argument due to its simplicity and clarity, I will turn to the case of

quantum mechanics to add more physical relevance.

2 QuantumMechanics

For more generality, I will consider n-dimensional tile space. Let each tile be rep-

resented by a member of Zn. From each tile, one can move along 2n directions,

characterized by n basis vectors ±e1,±e2, ...,±en (|ei |=1 for all i). Then, the follow-

ing claim is true (see Appendix B for the proof):

Isotropy-qm. For any quantum mechanical system α with initial po-

sition spread out in a sufficiently large region A ⊂ Zn that is approxi-

mately rotational invariant, for any y,z ∈ Zn that have approximately

the same Euclidean distance to A, the amplitudes of α at y and z are

approximately the same.9

This implies that for any two regions that consist of tiles with similar Euclidean

distances to the starting region, the probability of α showing up in those regions

are approximately the same. Assuming that such probabilities are our only observ-

9“Spread out” in a region implies the position space wavefunction of the system does not varymuch in short distances. The Euclidean distance between a tile and a region is equal to that betweenthe tile and a center of the region (i.e., a tile that minimizes the maximal Euclidean distance to othertiles in the region).

The requirement that the starting region is rotational invariant may sound fishy, but the idea is toassume that our preparation and observation of a suitably localized wavefunction should be them-selves approximately rotational invariant (that is, they do not significantly depend on in whichdirections we prepare or observe them). If this condition cannot be satisfied, then this alreadyconstitutes a failure of observational isotropy in quantum mechanics (although this is possible, adiscussion of it is beyond the current scope because it involves experimental details).

10

ables, there is an embedding from the tile space to Euclidean space that preserves

all structures and observations, as in the previous case.

Of course, while this case may have more physical relevance than random walk,

this is still a very simplistic toy example. For one thing, quantum mechanics is not

a fundamental theory due to its conflict with relativity (in this example, the tempo-

ral dimension is still pre-relativistically separate from the spatial dimensions and

is not discretized). Quantum field theory is more fundamental, and there is indeed

a sentiment in the community that large-scale isotropy will emerge from funda-

mental laws defined over discrete spacetime.10 But to convert such a sentiment

into rigorous frameworks, theorems, and proofs, there are still a lot to be done

and nothing certain can be said at this stage. Furthermore, even quantum field

theory will not be realistically enough for our purposes, since we have not solved

the problem of quantum gravity (that is, incorporating the gravitational field into

the frameworks of quantum field theory). The successful handling of quantum

gravity is important to a realistic story concerning the fundamental structure of

spacetime.

One may wonder at this point why the isotropic amplitudes in this case or the

isotropic probabilities in the previous case mean the emergence of Euclidean ge-

ometry. Strictly speaking, observable Euclidean geometry is not yet fully recovered

since the apparent distances are not even defined. But this is not a problem in prin-

ciple: all matters, including length-measuring devices, are reducible to quantum

fields or something more fundamental. Thus, ensuring the observable quantities

in the fundamental theory to be isotropic also ensures that the apparent distances

will be Euclidean.10This is a common hope or even belief, I think, in the community of lattice quantum field

theory (LQFT). Note that the lattice involved in LQFT may not be interpreted realistically, but as acomputational device. (For example, see Montvay and Munster 1994)

11

3 Where Weyl got wrong (or: against geometricism)

I have laid out the main results and my objection to Weyl’s argument, but it is

worth expounding on where Weyl went wrong. In particular, I will criticize two

related assumptions: (1) large-scale or observable physical geometry is determined

by fundamental spacetime structures independent of dynamic laws; (2) some ge-

ometric structures (including the metric structure) are ontologically and explana-

torily prior to dynamics and must be presumed by the latter. These may be called

“two tenets of geometricism,” the view that geometry is more fundamental than

dynamics, with the opposite view called dynamicism.

First, the Weyl tile argument mistakenly assumes how large-scale distances

emerge from the fundamental structures. It assumes a counting account of dis-

tance in the tile space:

Distance-by-counting. For any tiles A,B, if C1,C2, ...Cn are the least

number of tiles that are pairwise connected and connected to A,B, then

the distance between A,B is n+ 1.

Given (1), this account of distance is perfectly intuitive and natural for discrete

space and has been endorsed by many, including Riemann in his foundational

work (1866) for differential geometry. Apriorily, this seems the best account of

distance due to its simplicity and elegance (see Forrest 1995). However, it misses

the natural possibility that the distances that we observe (or physical geometry in

general) cannot be determined independently from the dynamic laws. In my cases,

dynamics play a crucial rule in determining physical geometry. The observable

istropy is determined by the dynamic laws that govern the movement of the tiny

cat in the case of random walk and the evolution of the wavefunction in the case

of quantum mechanics. More generally, since we observe physical geometry with

various measuring devices, it is natural to expect that the measurement we get is

12

partly determined by how those devices work. (But this is a hotly debated claim

in the debate between geometricism and dynamicism that I discuss elsewhere; see

Brown 2005, Maudlin 2012, Norton 2008, [Anonymized]).

Second, it is a mistake to assume that geometry must be presumed by dynam-

ics. The difference between this assumption and the previous one is this: one may

grant that the apparent geometry arises from dynamic laws but still insist that

some fundamental geometric structure must be presumed by dynamics. For exam-

ple, in the famous Poincare disk scenario, the geometry appears hyperbolic to the

residents of the disk because of the dynamics (there is a universal force that shrinks

rigid rods and bends light beams). But we know by stipulation that such dynamics

laws are still defined over Euclidean geometry, which exists fundamentally. Thus

the fact that dynamic laws play a role in determining apparent geometry does not

necessarily mean there is no underlying “real” geometry. It is a common belief

that dynamics laws need to be “written on” spacetime geometry (Earman 1989).

But the two cases I presented have shown clearly that dynamic laws do not need

any metric structure. For example, in articulating the Schrodinger equation, I

have only appealed to the domain of all tiles and arguably the spatial topology

expressed by connectedness. One may argue that the set of tiles (and the topology)

should count as geometrical structures, and therefore I haven’t refuted the second

assumption. Fair enough—in this paper, I only focus on arguing that no metric

structure needed to be presumed by the dynamics. To get rid of spacetime alto-

gether, we will need a very different framework such as “spacetime algebraicism”

rather than the standard point-set-theoretic framework, which I will not delve into

here (e.g., see Geroch 1972, Connes 2013, Chen and Fritz, 2021).11

Note that in the two cases, even if there is a metric structure captured by

Distance-by-Counting, or any number of ways, it has no empirical consequences,

11These authors did not discuss the discrete case (explicitly). But the discrete case can be a specialcase of the formalism proposed in (Chen and Fritz 2021).

13

since it does not play any role in the dynamic laws. This is, then, a considera-

tion against positing such a structure, since we should not posit structures with no

empirical consequences.

4 Conclusion

Weyl’s argument against discrete space implicitly assumes that the symmetries of

geometry help determine symmetries of matter systems. It follows that since the

geometry of the tile space is not rotational invariant, the physical laws and observ-

ables will also typically lack rotational invariance. However, I have shown that the

observable physical states in the cases of random walk and quantum mechanics

are decoupled from the tile geometry. Even though the tile space radically violates

rotational invariance in the sense that the diagonal of a square contains twice as

many steps as the side, the observables are still approximately rotational invariant.

14

A RandomWalk

Theorem A.1 The probability of reaching (x,y) from (0,0) after n steps is equal to

(1/4)n(

n(n+ x+ y)/2

)(n

(n+ x − y)/2

)(4)

Proof We imagine that in every step the cat takes, the cat actually takes two

half-steps along the diagonal directions. We assume that there are four possi-

ble ways the cat can move in one step from position 0: (1/2,1/2) + (1/2,−1/2),

(1/2,1/2)+(−1/2,1/2), (−1/2,−1/2)+(1/2,−1/2), (−1/2,−1/2)+(−1/2,1/2). This cor-

responds to the four possible ways to move in the original situation (±1,±1). We

can consider the original situation and the imagined situation as two ways of rep-

resenting the same physical situation. As an analogy, in chess, a knight’s move

can be equivalently considered as one L-shaped step or as consisting of first mov-

ing one row (or file) and then moving two files (or rows). Let’s call the number

of steps that the cat takes in the aforementioned four possible ways respectively

ac,ad,bc,bd (Figure 3). Because the number of steps involving a (i.e., ac + ad) and

Figure 3

the number of steps involving c (i.e., ac + bc) are independent, the probability of

15

reaching (x,y) after n steps is equal to

(1/2)n(

nac+ ad

)· (1/2)n

(n

ac+ bc

)

with ac+ ad + bc+ bd = n and ac− bd = x,ad − bc = y. It follows that the probability

of reaching (x,y) after n steps is equal to

(1/4)n(

n(n+ x+ y)/2

)(n

(n+ x − y)/2

)

Theorem A.2 Assuming x� n and y� n, (4) is approximately equal to

2e(x2+y2)/n

π√n2 − (x2 + y2)

(5)

Proof Assuming both n and n− k are very large, we have the following theorem

(Gallager 1968):

(nk

)→

√n

2πk(n− k)enh(k/n) (6)

where k ∈ [1,n − 1]∩Z and h = −x lnx − (1 − x) ln(1 − x). Plugging (4) into (6), the

result would be of the form αeβ , where

α = (1/4)n√

4n2

π2(n4 −n2(x2 + y2) + (x2 − y2)2)(7)

Since x� n and y � n, we have (x2 − y2)2 � n2, and thus (7) is approximately

equal to

(1/4)n√

4n2

π2n4 −n2(x2 + y2)= (1/4)n

2

π√n2 − (x2 + y2)

(8)

Note that this expression involve only the Euclidean form x2 + y2, which is rota-

16

tional invariant. (This is the first approximation involved in the proof.)

The exponent β is nh(n+x+y2n ) +nh(n+x−y

2n ). Let u = x/2n and v = y/2n. Then:

β = nh(12

+u + v) +nh(12

+u − v) (9)

Since u + v and u − v are close to zero, we can approximate h(1/2 + u + v) and

h(12 +u − v) by expanding h around 1/2 up to the second order

h(12

+ p) = h(12

) + h′(12

)p+ 1/2h′′(12

)p2 (10)

where

h(12

) = −1/2ln1/2− 1/2ln1/2 = ln2

h′(12

) = − ln1/2 + ln1/2 = 0

h′′(12

) = −2− 2 = 4

This is the second and last approximation involved in the proof. Plugging them

into (9), we obtain:

β ≈ n(ln2− 2(u + v)2) +n(ln2− 2(u − v)2) (11)

= n ln4− 4n(u2 + v2) = n ln4−x2 + y2

n(12)

Then, the exponential eβ is equal to en ln4− x2+y2

n = 4ne(x2+y2)/n. Together with (8), we

obtain that αeβ is equal to (5). �

17

B QuantumMechanics

Theorem B.1 For any quantum mechanical system α with initial position wavefunc-

tion spread out in a sufficiently large region A ⊂ Zn that is approximately rotational

invariant, for any y,z ∈ Zn that have approximately the same Euclidean distance to A,

the amplitudes of α at y and z are approximately the same.

Proof We start from the Schrodinger equation, which governs the dynamics of

quantum mechanical systems (setting the Planck constant ~ to one):

iddt

Ψ (t) = HΨ (t),

where Ψ (t) is the wave function of the system (complex-valued probability ampli-

tude) and H is the Hamiltonian operator on the wavefunction (total energy of the

system). Clearly, we first need to formulate the discrete version of the Schrodinger

equation. I will use the same notations for the discrete case. First, the discrete

version of the wavefunction Ψ (t) for any given t is a complex-valued function over

the tile space:

Ψ (t) : Zn→ C.

For the right side, we need to discretize the Hamiltonian. If we set the mass to

one and ignore the potential energy, the Hamiltonian is equal to its kinetic energy

−12

∑i

(∂

∂xi)2. An obvious discrete definition of ∂

∂xiwould be this:

∂

∂xiΨ (t,x) = Ψ (t,x+ ei)−Ψ (t,x), or

= Ψ (t,x)−Ψ (t,x − ei)

18

Intuitively, in the one dimensional case, the derivative of a function at a point (i.e.,

tile) is the difference between the value of the function at the point and the value at

its neighboring point of choice. In the following, I will continue using the notation

“ ∂∂xi

” with this discrete definition. Then, the discrete version of the Hamiltonian

is as follows (concerning only its kinetic energy part):12

HΨ (t,x) = −12

∑i

(Ψ (t,x+ ei) +Ψ (t,x − ei)− 2Ψ (t,x))

The Fourier series of Ψ (t) is its momentum space counterpart Ψ (t) : Rn→ C:

Ψ (t,p) =∑x∈Zn

e−2πipxΨ (t,x)

where “px” refers to the inner product of n-vectors p and x. Notice that this func-

tion is periodical. That is, Ψ (t,p) = Ψ (t,p + ei).13 Therefore, we can consider

Ψ (t) as a complex-valued function defined on the quotient space Rn/Zn, which

means that we “collapse” all real space points with integer distances away into

one point. This domain can be represented by the unit square around the ori-

gin: B = [−1/2,1/2] × [−1/2,1/2] (“B” stands for “Brillouin zone”). Then we have

Ψ (t) : B→ C. We can transform it back to the position space wavefunction in the

following way:

Ψ (t,x) =∫p∈B

e2πipxΨ (t,p)dp

12This is obtained by applying ∂∂xi

twice, in opposite directions. If it is applied in the samedirection, we would have

HΨ (t,x) = −12

∑i

(Ψ (t,x+ 2ei)− 2Ψ (t,x+ ei) +Ψ (t,x))

which is also a legitimate choice, but makes calculation slightly more complicated.13Here’s the derivation (omitting t for the moment):

Ψ (p+ ei) =∑x∈Zn

e−2πi(p+ei )xΨ (x) =∑x∈Zn

e−2πipxe−2πieiΨ (x) =∑x∈Zn

e−2πipxΨ (x) = Ψ (p)

19

By applying fourier series transform to HΨ (t,x), we can get the momentum

space Hamiltonian H with i ddt Ψ (t) = HΨ (t), which is

HΨ (t,p) = −12

∑i

(e2πipi + e−2πipi − 2)Ψ (t,p),

where pi = p · ei , namely the i-th component of p.14 Let’s abbreviate the expression

“−12∑i(e

2πipi +e−2πipi −2)” as “Cow(p)”. Then, HΨ (t,p) = CowΨ (t,p). Now we can

solve the Schrodinger equation in the momentum space:

Ψ (t,p) = (e−iHtΨ )(p) = e−iCow(p)tΨ (0,p)

Recall that we assume the quantum mechanical system under consideration starts

with a wavefunction sufficiently spread out around origin 0 at t = 0. Then, Ψ (0,p)

is negligibly small for large |p| and only nonnegligible for small |p|.15 We can apply

the inverse Fourier series transform to the above result and obtain:

Ψ (t,x) =∫p∈B

e2πipxe−iCow(p)tΨ (0,p)dp

We can show that Cow(p) is approximately rotational invariant (a spherical

14To get this result, we observe that Ψ (t,x + ei) can be transformed to e2πipi Ψ (t,p) through thefollowing step. The rest is similar.

Ψ (t,x+ ei)→∑x∈Zn

e−2πipxΨ (t,x+ ei)

=∑x∈Zn

e−2πip(x−ei )Ψ (t,x)

=∑x∈Zn

e−2πipxe2πipeiΨ (t,x)

= e2πipi Ψ (t,p)

15Given Ψ (t,p) =∑x∈Zn e

−2πipxΨ (t,x) and Ψ (0,x) does not vary much over small distances, whenp is large, e−2πipxΨ (t,x) tend to cancel off over small variation of x, so the results add up small(“destructive interference”). When p is very small (i.e.,p � 1), then there is no such destructiveinterference, and therefore the sum is much more significant.

20

cow) when p is sufficiently small:

Cow(p) = −12

∑i

(e−2πipi + e2πipi − 2)

= −∑i

(cos(2πpi)− 1)

= −∑i

(1− 2π2p2i − 1 +O(p4

i ))

= 2π2∑i

p2i −O(p4)

Here∑i p

2i is the square of the Euclidean length of p in the momentum space and is

rotational invariant. O(p4) is much smaller and can be ignored if |p| is sufficiently

small. Now if we look at the whole integral, we can see that large values of |p| does

not contribute much to the integral because Ψ (0,p) is negligibly small for large |p|.

So only small values of |p| make main contribution to the integral. Since we can

ignoreO(p4) when |p| is sufficiently small, then the whole integral is approximately

invariant under any rotation of p in Cow.

It follows that Ψ (t,x) is also approximately rotational invariant. To see this, we

first extend Ψ (t,x) to a continuous function over Rn by simply plug in continuous

values for x in the above integral (note that the extension is merely heuristic). Let

A be a Euclidean rotation matrix. We have:

Ψ (t,Ax) =∫p∈B

e2πipAxe−iCow(p)tΨ (0,p)dp

=∫p∈AB

e2πiApAxe−iCow(Ap)tΨ (0,Ap)dp (substitute dp by dA−1p)

=∫p∈AB

e2πipxe−iCow(Ap)tΨ (0,p)dp (ApAx = px; Ψ (0,Ap) = Ψ (0,p)16)

≈∫p∈B

e2πipxe−iCow(p)tΨ (0,p)dp17 (Cow(Ap) ≈ Cow(p))

= Ψ (t,x)

21

This means that for any two points a,b inRn that have the same Euclidean distance

to the origin, Ψ (t,a) and Ψ (t,b) have approximately the same value. Of course, x,y

may not be on the lattice. But their nearest lattice points have very similar Ψ -

values, since changing x in the above integral by one (when |x| is very large) has

very small effect on the result. Therefore, we can conclude that for any two lattice

points that have approximately the same Euclidean distance to the origin, Ψ (t) has

approximately the same value on them. This can be straightforwardly generalized

to any lattice point as the starting position (as it is just a matter of coordinate

translation). It follows that Theorem B.1 is true. �

13Ψ (0,p) is approximately rotational invariant because Ψ (0,x) is approximately rotational in-variant by stipulation in Theorem B.1.

17Integrating over a rotated Brillouin zone AB is approximately the same as integrating over theBrillouin zone B because only the small p around the origin make main contribution to the integral.

22

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