Feynman checkers:
towards algorithmic quantum theory

The simplest way to understand what is the model about is the classical double-slit experiment (see Figure 2). In this experiment, a (coherent) beam of electrons is directed towards a plate pierced by two parallel slits, and the part of the beam passing through the slits is observed on a screen behind the plate. If one of the slits is closed, then the beam illuminates a spot on the screen. If both slits are open, one would expect a larger spot, but in fact one observes a sequence of bright and dark bands (interferogram).

This shows that electrons behave like a wave: the waves travel through both slits, and the contributions of the two paths either amplify or cancel each other depending on the final phases.

Further, if the electrons are sent through the slits one at time, then single dots appear on the screen, as expected. Remarkably, however, the same interferogram with bright and dark bands emerges when the electrons are allowed to build up one by one. One cannot predict where a particular electron hits the screen; all we can do is to compute the probability to find the electron at a given place.

The Feynman sum-over-paths (or path integral) method of computing such probabilities is to assign phases to all possible paths and to sum up the resulting waves (see [11, 12]). Feynman checkers (or Feynman checkerboard) is a particularly simple combinatorial rule for those phases in the case of an electron freely moving (or better jumping) along a line.

The beginning. The checkers model was invented by R. Feynman in 1940s [43] and first published in 1965 [12]. In Problem 2.6 there, a function on a lattice of small step $\varepsilon$ was constructed (called kernel; see (2)) and the following task was posed:

If the time interval is very long ($t_{b}-t_{a}\gg\hbar/mc^{2}$) and the average velocity is small [$x_{b}-x_{a}\ll c(t_{b}-t_{a})$], show that the resulting kernel is approximately the same as that for a free particle [given in Eq. (3-3)], except for a factor $\exp[(imc^{2}/\hbar)(t_{b}-t_{a})]$.

Mathematically, this means that the kernel (divided by $2i\varepsilon\exp[(-imc^{2}/\hbar)(t_{b}-t_{a})]$) asymptotically equals free-particle kernel (24) (this is Eq. (3-3) from [12]) in the triple limit when time tends to infinity, whereas the average velocity and the lattice step tend to zero (see Table 1 and Figure 3). Both scaling by the lattice step and tending it to zero were understood, otherwise the mentioned ‘‘exceptional’’ factor would be different (see Example 4). We show that the assertion, although incorrect literally, holds under mild assumptions (see Corollary 5).

Although the Feynman problem might seem self-evident for theoretical physicists, even the first step of a mathematical solution (disproving the assertion as stated) is not found in literature. As usual, the difficulty is to prove the convergence rather than to guess the limit.

In 1972 J. Narlikar discovered that the above kernel reproduces the spin-$1/2$ retarded propagator in the different limit when the lattice step tends to zero but time stays fixed [36] (see Table 1, Figures 4 and 1, Corollary 6). In 1984 T.Jacobson–L.Schulman repeated this derivation, applied stationary phase method among other bright ideas, and found the probability of changing the movement direction [23] (cf. Theorem 6). The remarkable works of that period contain no mathematical proofs, but approximate computations without estimating the error.

Ising model. In 1981 H. Gersch noticed that Feynman checkers can be viewed as a $1$-dimensional Ising model with imaginary temperature or edge weights (see §2.2 and [18], [23, §3]). Imaginary values of these quantities are usual in physics (e.g., in quantum field theory or in alternating current networks). Due to the imaginarity, contributions of most configurations cancel each other, which makes the model highly nontrivial in spite of being $1$-dimensional. In particular, the model exhibits a phase transition (see Figures 1 and 5). Surprisingly, the latter seems to have never been reported before. Phase transitions were studied only in more complicated $1$-dimensional Ising models [25, §III], [35], in spite of a known equivalent result, which we are going to discuss now (see Theorem 1(B) and Corollary 3).

Quantum walks. In 2001 A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous performed a breakthrough [1]. They studied Feynman checkers under names one-dimensional quantum walk and Hadamard walk; although those cleverly defined models were completely equivalent to Feynman’s simple one. They computed the large-time limit of the model (see Theorem 2). They discovered several striking properties having sharp contrast with both continuum theory and the classical random walks. First, the most probable average electron velocity in the model equals $1/\sqrt{2}$ of the speed of light, and the probability of exceeding this value is very small (see Figures 5 and 1 to the left and Theorem 1(B)). Second, if an absorbing boundary is put immediately to the left of the starting position, then the probability that the electron is absorbed is $2/\pi$. Third, if an additional absorbing boundary is put at location $x>0$, the probability that the electron is absorbed to the left actually increases, approaching $1/\sqrt{2}$ in the limit $x\to+\infty$. Recall that in the classical case both absorption probabilities are $1$. In addition, they found many combinatorial identities and expressed the above kernel through the values of Jacobi polynomials at a particular point (see Remark 3; cf. [48, §2]).

N. Konno studied a biased quantum walk [29, 30], which is still essentially equivalent to Feynman checkers (see Remark 4). He found the distribution of the electron position in the (weak) large-time limit (see Figure 5 and Theorem 1(B)). This result was proved mathematically by G.R. Grimmett–S. Janson–P.F. Scudo [20]. In their seminal paper [47] from 2012, T. Sunada–T. Tate found and proved a large-time asymptotic formula for the distribution (see Theorems 2–4). This was a powerful result but it still could not solve the Feynman problem because the error estimate was not uniform in the lattice step. In 2018 M. Maeda et al. proved a bound for the maximum of the distribution for large time [33, Theorem 2.1].

Quantum walks were generalized to arbitrary graphs and applied to quantum algorithms (see Figure 6 and [17] for an implementation). We refer to the surveys by M.J. Cantero–F.A. Grünbaum–L. Moral–L. Velázquez, N. Konno, J. Kempe, and S.E. Venegas-Andraca [7, 30, 27, 50, 31] for further details in this direction.

Lattice quantum field theories. In a more general context, this is a direction towards creation of Minkowskian lattice quantum field theory, with both space and time being discrete [2]. In 1970s F. Wegner and K. Wilson introduced lattice gauge theory as a computational tool for gauge theory describing all known interactions (except gravity); see [34] for a popular-science introduction. This culminated in determining the proton mass theoretically with error less than $2\%$ in a sense. This theory is Euclidean in the sense that it involves imaginary time. Likewise, an asymptotic formula for the propagator for the (massless) Euclidean lattice Dirac equation [28, Theorem 4.3] played a crucial role in the continuum limit of the Ising model performed by D. Chelkak–S. Smirnov [8]. Similarly, asymptotic formulae for the Minkowskian one (Theorems 2–5) can be useful for missing Minkowskian lattice quantum field theory. Several authors argue that Feynman checkers have the advantage of no fermion doubling inherent in Euclidean lattice theories and avoid the Nielsen–Ninomiya no-go theorem [4, 15].

Several upgrades of Feynman checkers have been discussed. For instance, around 1990s B. Gaveau–L. Schulman and G. Ord added electromagnetic field to the model [16, 38]. That time they achieved neither exact charge conservation nor generalization to non-Abelian gauge fields; this is fixed in Definition 3. Another example is adding mass matrix by P. Jizba [24].

It was an old dream to incorporate also checker paths turning backwards in time or forming cycles [43, p. 481–483], [22]; this would mean creation of electron-positron pairs, celebrating a passage from quantum mechanics to quantum field theory. One looked for a combinatorial model reproducing the Feynman propagator rather than the retarded one in the continuum limit (see Table 1). Known constructions (such as hopping expansion) did not lead to the Feynman propagator because of certain lattice artifacts (e.g., the title of [39] is misleading: the Feynman propagator is not discussed there). In the massless case, a noncombinatorial construction of Feynman propagator on the lattice was provided by C. Bender–L. Mead–K. Milton–D. Sharp in [2, §9F] and [3, §IV]. In Definition 6 the desired combinatorial construction is finally given.

Another long-standing open problem is to generalize the model to the $4$-dimensional real world. In his Nobel prize lecture, R. Feynman mentioned his own unsuccessful attempts. There are several recent approaches, e.g., by B. Foster–T. Jacobson from 2017 [15]. Those are not yet as simple and beautiful as the original $2$-dimensional model, as it is written in [15, §7.1] itself.

On physical and mathematical works. The physical literature on the subject is quite extensive [50], and we cannot mention all remarkable works in this brief overview. Surprisingly, in the whole literature we have not found the shouting property of concentration of measure for lattice step tending to zero (see Corollary 7). Many papers are well-written, insomuch that the physical theorems and proofs there could be carelessly taken for mathematical ones (see the end of §12). We are aware of just a few mathematical works on the subject, such as [20, 47, 33].

We solve mathematically a problem by R. Feynman from 1965, which was to prove that his model is consistent with the continuum one, namely, reproduces the usual quantum-mechanical free-particle kernel for large time, small average velocity, and small lattice step (see Corollary 5). We compute large-time and small-lattice-step asymptotic formulae for the lattice propagator, uniform in the model parameters (see Theorems 2 and 5). For the first time we observe and prove concentration of measure in the continuum limit: the average velocity of an electron emitted by a point source is close to the speed of light with high probability (see Corollary 7). The results can be interpreted as asymptotic properties of Young diagrams (see Corollary 2) and Jacobi polynomials (see Remark 3).

All these results are proved mathematically for the first time. For their statements, just Definition 2 is sufficient. In Definitions 3 and 6 we perform a coupling to lattice gauge theory and the second quantization of the model, promoting Feynman checkers to a full-fledged lattice quantum field theory.

First we give the definitions and precise statements of the results, and in the process provide a zero-knowledge examples for basic concepts of quantum theory. These are precisely those examples that R. Feynman presents first in his own books: Feynman checkers (see §3) is the first specific example in the whole book [12]. The thin-film reflection (see §7) is the first example in [11]; see Figures 10–11 there. Thus we hope that these examples could be enlightening to readers unfamiliar with quantum theory.

We start with the simplest (and rough) particular case of the model and upgrade it step by step in each subsequent section. Before each upgrade, we summarize which physical question does it address, which simplifying assumptions does it resolve or impose additionally, and which experimental or theoretical results does it reproduce. Some upgrades (§§7–9) are just announced to be discussed in a subsequent publication. Our aim is (1+1)-dimensional lattice quantum electrodynamics (‘‘QED’’) but the last step on this way (mentioned in §10) is still undone. Open problems are collected in §11. For easier navigation, we present the upgrades-dependence chart:

Hopefully this is a promising path to making quantum field theory rigorous and algorithmic. An algorithmic quantum field theory would be a one which, given an experimentally observable quantity and a number $\delta>0$, would provide a precise statement of an algorithm predicting a value for the quantity within accuracy $\delta$. (Surely, the predicted value needs not to agree with the experiment for $\delta$ less than accuracy of theory itself.) See Algorithm 1 for a toy example. This would be an extension of constructive quantum field theory (currently far from being algorithmic). Application of quantum theory to computer science is in mainstream now, but the opposite direction could provide benefit as well. (Re)thinking algorithmically is a way to make a subject available to nonspecialists, as it is happening with, say, algebraic topology.

The paper is written in a mathematical level of rigor, in the sense that all the definitions, conventions, and theorems (including corollaries, propositions, lemmas) should be understood literally. Theorems remain true, even if cut out from the text. The proofs of theorems use the statements but not the proofs of the other ones. Most statements are much less technical than the proofs; hence the proofs are kept in a separate section (§12) and long computations are kept in [44]. In the process of the proofs, we give a zero-knowledge introduction to the main tools to study the model: combinatorial identities, the Fourier transform, the method of moments, the stationary phase method. Remarks are informal and usually not used elsewhere (hence skippable). Text outside definitions, theorems, proofs is neither used formally.

We first give an informal definition of the model in the spirit of [11] and then a precise one.

On an infinite checkerboard, a checker moves to the diagonal-neighboring squares, either upwards-right or upwards-left (see Figure 7 to the left). To each path $s$ of the checker, assign a vector ${a}(s)$ as follows (see Figure 7 to the middle). Take a stopwatch that can time the checker as it moves. Initially the stopwatch hand points upwards. While the checker moves straightly, the hand does not rotate, but each time when the checker changes the direction, the hand rotates through $90^{\circ}$ clockwise (independently of the direction the checker turns). The final direction of the hand is the direction of the required vector ${a}(s)$. The length of the vector is set to be $1/2^{(t-1)/2}$, where $t$ is the total number of moves (this is just a normalization). For instance, for the path $s$ in Figure 7 to the top-middle, the vector ${a}(s)=(0,-1/2)$.

Denote by ${a}(x,t):=\sum_{s}{a}(s)$ the sum over all the checker paths from the square $(0,0)$ to the square $(x,t)$, starting with the upwards-right move. For instance, ${a}(1,3)=(0,-1/2)+(1/2,0)=(1/2,-1/2)$; see Figure 7 to the right. The length square of the vector ${a}(x,t)$ is called the probability to find an electron in the square $(x,t)$, if it was emitted from $(0,0)$ (see §2.2 for a discussion of the terminology). The vector ${a}(x,t)$ itself is called the arrow [11, Figure 6].

Let us summarize this construction rigorously.

Figure 8 and Table 2 depict the arrows $a(x,t)$ and the probabilities $P(x,t)$ for small $x,t$. Figure 9 depicts the graphs of $P(x,1000)$, $a_{1}(x,1000)$, and $a_{2}(x,1000)$ as functions in an even number $x$. We see that under variation of the final position $x$ at a fixed large time $t$, right after the peak the probability falls to very small although still nonzero values. What is particularly interesting is the unexpected position of the peak, far from $x=1000$. In Figure 1 to the left, the color of a point $(x,t)$ with even $x+t$ depicts the value $P(x,t)$. Notice that the sides of the apparent angle are not the lines $t=\pm x$ but the lines $t=\pm\sqrt{2}x$ (see Theorem 1(A)).

Let us comment on the physical interpretation of the model and ensure that it captures unbelievable behavior of electrons. There are two different interpretations; see Table 3.

Standard interpretation. Here the $x$- and $t$-coordinates are interpreted as the electron position and time respectively. Sometimes (e.g., in Example 1) we a bit informally interpret both as position, and assume that the electron performs a ‘‘uniform classical motion’’ along the $t$-axis. We work in the natural system of units, where the speed of light, the Plank and the Boltzmann constants equal $1$. Thus the lines $x=\pm t$ represent motion with the speed of light. Any checker path lies above both lines, i.e. in the light cone, which means agreement with relativity: the speed of electron cannot exceed the speed of light.

To think of $P(x,t)$ as a probability, consider the $t$-coordinate as fixed, and the squares $(-t,t),(-t+2,t),\dots,(t,t)$ as all the possible outcomes of an experiment. For instance, the $t$-th horizontal might be a screen detecting the electron. We shall see that all the numbers $P(x,t)$ on one horizontal sum up to $1$ (Proposition 2), thus indeed can be considered as probabilities. Notice that the probability to find the electron in a set $X\subset\mathbb{Z}$ is $P(X,t):=\sum_{x\in X}P(x,t)=\sum_{x\in X}|a(x,t)|^{2}$ rather than $\left|\sum_{x\in X}a(x,t)\right|^{2}$ (cf. [11, Figure 50]).

In reality, one cannot measure the electron position exactly. A fundamental limitation is the electron reduced Compton wavelength $\lambda=1/m\approx 4\cdot 10^{-13}$ meters, where $m$ is the electron mass. Physically, the basic model approximates the continuum by a lattice of step exactly $\lambda$. But that is still a rough approximation: one needs even smaller step to prevent accumulation of approximation error at larger distances and times. For instance, Figures 1 and 10 to the left show a finite-lattice-step effect (renormalization of speed of light): the average velocity $x/t$ cannot exceed $1/\sqrt{2}$ of the speed of light with high probability. (An explanation in physical terms: lattice regularization cuts off distances smaller than the lattice step, hence small wavelengths, hence large momenta, and hence large velocities.) A more precise model is given in §3: compare the plots in Figure 1.

As we shall see now, the model qualitatively captures unbelievable behavior of electrons. (For correct quantitative results like exact shape of an interferogram, an upgrade involving a coherent source is required; see §6.)

The probability to find an electron in the square $(x,t)$ subject to absorption in a subset $B\subset\mathbb{Z}^{2}$ is defined analogously to $P(x,t)$, only the summation is over checker paths $s$ that have no common points with $B$ possibly except $(0,0)$ and $(x,t)$. The probability is denoted by $P(x,t\text{ bypass }B)$. Informally, this means an additional outcome of the experiment: the electron has been absorbed and has not reached the  screen. In the following example, we view the two black squares $(\pm 1,1)$ on the horizontal $t=1$ as two slits in a horizontal plate (cf. Figure 2).

The standard interpretation of Feynman checkers is also known as the Hadamard walk, or more generally, the 1-dimensional quantum walk or quantum lattice gas. Those are all equivalent but lead to generalizations in distinct directions [50, 31, 51]. For instance, a unification of the upgrades from §3–7 gives a general inhomogeneous 1-dimensional quantum walk.

The striking properties of quantum walks discussed in §1.2 are stated precisely as follows:

Recently M. Dmitriev [10] has found $\sum_{t=1}^{\infty}P(n,t\text{ bypass }\{x=n\})$ for a few values $n\neq 0$ (see Problem 6). Similar numbers appear in the simple random walk on $\mathbb{Z}^{2}$ [41, Table 2].

Spin-chain interpretation. There is a very different physical interpretation of the same model: a $1$-dimensional Ising model with imaginary temperature and fixed magnetization.

Recall that a configuration in the Ising model is a sequence $\sigma=(\sigma_{1},\dots,\sigma_{t})$ of $\pm 1$ of fixed length. The magnetization and the energy of the configuration are $\sum_{k=1}^{t}\sigma_{k}/t$ and $H(\sigma)=\sum_{k=1}^{t-1}(1-\sigma_{k}\sigma_{k+1})$ respectively. The probability of the configuration is $e^{-\beta H(\sigma)}/Z(\beta)$, where the inverse temperature $\beta=1/T>0$ is a parameter and the partition function $Z(\beta):=\sum_{\sigma}e^{-\beta H(\sigma)}$ is a normalization factor. Additional restrictions on configurations $\sigma$ are usually imposed.

Now, moving the checker along a path $s$, write ‘‘$+$’’ for each upwards-right move, and ‘‘$-$’’ for each upwards-left one; see Figure 11 to the left. The resulting sequence of signs is a configuration in the Ising model, the number of turns in $s$ is one half of the configuration energy, and the ratio of the final $x$- and $t$-coordinates is the magnetization. Thus $a(x,t)=\sum_{s}a(s)$ coincides (up to a factor not depending on $x$) with the partition function for the Ising model at the imaginary inverse temperature $\beta=i\pi/4$ under the fixed magnetization $x/t$:

Notice a crucial difference of the resulting spin-chain interpretation from both the usual Ising model and the above standard interpretation. In the latter two models, the magnetization $x/t$ and the average velocity $x/t$ were random variables; now the magnetization $x/t$ (not to be confused with an external magnetic field) is an external condition. The configuration space in the spin-chain interpretation consists of sequences of ‘‘$+$’’ and ‘‘$-$’’ with fixed numbers of ‘‘$+$’’ and ‘‘$-$’’. Summation over configurations with different $x$ or $t$ would make no sense: e.g., recall that $P(X,t)=\sum_{x\in X}|a(x,t)|^{2}$ rather than $\left|\sum_{x\in X}a(x,t)\right|^{2}$.

Varying the magnetization $x/t$, viewed as an external condition, we observe a phase transition: (the imaginary part of) the limiting free energy density $-\log a(x,t)/\beta t$ is non-analytic when $x/t$ passes through $\pm 1/\sqrt{2}$ (see Figure 5 to the middle and Corollary 3). The phase transition emerges as $t\to\infty$. It is interesting to study other order parameters, for instance, the ‘‘probability’’ $i\,a_{2}(x,t)/a(x,t)$ of equal signs at the endpoints of the spin chain (see Figure 10 and Problems 4–5). These quantities are complex just because the temperature is imaginary.

(A comment for specialists: the phase transition is not related to accumulation of zeroes of the partition function in the plane of complex parameter $\beta$ as in [25, §III] and [35]. In our situation, $\beta=i\pi/4$ is fixed, the real parameter $x/t$ varies, and the partition function $a(x,t)$ has no zeroes at all [37, Theorem 1].)

Young-diagram interpretation. Our results have also a combinatorial interpretation.

The number of steps (or inner corners) in a Young diagram with $w$ columns of heights $x_{1},\dots,x_{w}$ is the number of elements in the set $\{x_{1},\dots,x_{w}\}$; see Figure 11 to the left. Then the value $2^{(h+w-1)/2}\,a_{1}(h-w,h+w)$ is the difference between the number of Young diagrams with an odd and an even number of steps, having exactly $w$ columns and $h$ rows.

Interesting behaviour starts already for $h=w$ (see Proposition 4). For $h=w$ even, the difference vanishes. For $h=w=2n+1$ odd, it is $(-1)^{n}\binom{2n}{n}$. Such $4$-periodicity roughly remains for $h$ close to $w$ [45, Theorem 2]. For fixed half-perimeter $h+w$, the difference slowly oscillates as $h/w$ increases, attains a peak at $h/w\approx 3+2\sqrt{2}$, and then harshly falls to very small values (see Corollary 2 and Theorems 2–4).

Similarly, $2^{(h+w-1)/2}a_{2}(h-w,h+w)$ is the difference between the number of Young diagrams with an even and an odd number of steps, having exactly $w$ columns and less than $h$ rows. The behaviour is similar. The upgrade in §3 is related to Stanley character polynomials [48, §2].

Discussion of the definition. Now compare Definition 1 with the ones in the literature.

The notation ‘‘$a$’’ comes from ‘‘arrow’’ and ‘‘probability amplitude’’; other names are ‘‘wavefunction’’, ‘‘kernel’’, ‘‘the Green function’’, ‘‘propagator’’. More traditional notations are ‘‘$\psi$’’, ‘‘$K$’’, ‘‘$G$’’, ‘‘$\Delta$’’, ‘‘$S$’’ depending on the context. We prefer a neutral one suitable for all contexts.

The factor of $i$ and the minus sign in the definition are irrelevant (and absent in the original definition [12, Problem 2.6]). They come from the ordinary starting direction and rotation direction of the stopwatch hand, and reduce the number of minus signs in what follows.

The normalization factor $2^{(1-t)/2}$ can be explained by analogy to the classical random walk. If the checker were performing just a random walk, choosing one of the two possible directions at each step (after the obligatory first upwards-right move), then $|a(s)|^{2}$ $=2^{1-t}$ would be the probability of a path $s$. This analogy should be taken with a grain of salt: in quantum theory, the ‘‘probability of a path’’ has absolutely no sense (recall Example 1). The reason is that the path is not something one can measure: a measurement of the electron position at one moment strongly affects the motion for all later moments.

Conceptually, one should also fix the direction of the last move of the path $s$ (see [12, bottom of p.35]). Luckily, this is not required in the present paper (and thus is not done), but becomes crucial in further upgrades (see §4 for an explanation).

One could ask where does the definition come from. Following Feynman, we do not try to explain or ‘‘derive’’ it physically. This quantum model cannot be obtained from a classical one by the standard Feynman sum-over-paths approach: there is simply no clear classical analogue of a spin $1/2$ particle (cf. §4) and no true understanding of spin. ‘‘Derivations’’ in [1, 4, 36] appeal to much more complicated notions than the model itself. The true motivation for the model is its simplicity, consistency with basic principles (like probability conservation), and agreement with experiment (which here means the correct continuum limit; see Corollary 6).

Let us state several well-known basic properties of the model. The proofs are given in §12.1.

First, the arrow coordinates $a_{1}(x,t)$ and $a_{2}(x,t)$ satisfy the following recurrence relation.

This mimics the $(1+1)$-dimensional Dirac equation in the Weyl basis [42, (19.4) and (3.31)]

only the derivatives are replaced by finite differences, $m$ is set to $1$, and the normalization factor $1/\sqrt{2}$ is added. For the upgrade in §3, this analogy becomes transparent (see Remark 2). The Weyl basis is not unique, thus there are several forms of equation (1); cf. [23, (1)].

The Dirac equation implies the conservation of probability.

For $a_{1}(x,t)$ and $a_{2}(x,t)$, there is an ‘‘explicit’’ formula (more ones are given in Appendix A).

The following proposition is a straightforward corollary of the ‘‘explicit’’ formula.

In §3.1 we give more identities. The sequences $2^{(t-1)/2}a_{1}(x,t)$ and $2^{(t-1)/2}a_{2}(x,t)$ are present in the on-line encyclopedia of integer sequences [46, A098593 and A104967].

The following remarkable result was observed in [1, §4] (see Figures 5 and 10), stated precisely and derived heuristically in [29, Theorem 1], and proved mathematically in [20, Theorem 1]. See a short exposition of the latter proof in §12.2, and generalizations in §3.2.

Theorem 1(B) demonstrates a phase transition in Feynman checkers, if interpreted as an Ising model at imaginary temperature and fixed magnetization. Recall that then the magnetization $v$ is an external condition (rather than a random variable) and $P(\lceil vt\rceil,t)$ is the norm square of the partition function (rather than a probability). The distributional limit of $tP(\lceil vt\rceil,t)$ is discontinuous at $v=\pm 1/\sqrt{2}$, reflecting a phase transition (cf. Corollary 3).

One interprets $P(x,t,m,\varepsilon)$ as the probability to find an electron of mass $m$ in the square $\varepsilon\times\varepsilon$ with the center $(x,t)$, if the electron was emitted from the origin. Notice that the value $m\varepsilon$, hence $P(x,t,m,\varepsilon)$, is dimensionless in the natural units, where $\hbar=c=1$. In Figure 1 to the middle, the color of a point $(x,t)$ depicts the value $P(x,t,1,0.5)$ (if $x+t$ is an integer). Table 4 shows $a(x,t,m,\varepsilon)$ for small $x$ and $t$. Recently I. Novikov elegantly proved that the probability vanishes nowhere inside the light cone: $P(x,t,m,\varepsilon)\neq 0$ for $m>0$, $|x|<t$ and even $(x+t)/\varepsilon$ [37, Theorem 1].

Let us list known combinatorial properties of the model [50, 30]; see §12.1 for simple proofs.

This equation reproduces the Klein–Gordon equation $\tfrac{\partial^{2}a}{\partial t^{2}}-\tfrac{\partial^{2}a}{\partial x^{2}}+m^{2}a=0$ in the limit $\varepsilon\to 0$.

Informally, Huygens’ principle means that each black square $(x^{\prime},t^{\prime})$ on one horizontal acts like an independent point source, with the amplitude and phase determined by $a(x^{\prime},t^{\prime},m,\varepsilon)$.