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International Journal of Theoretical Physics, VoL 21, Nos. 6/7, 1982

Simulating Physics with Computers

Richard P. Feynman

Department of Physics, California Institute of Technology, Pasadena, California 91107

Received May 7, 1981

1. INTRODUCTION

On the program it says this is a keynote speech--and I don't know

what a keynote speech is. I do not intend in any way to suggest what should

be in this meeting as a keynote of the subjects or anything like that. I have

my own things to say and to talk about and there's no implication that

anybody needs to talk about the same thing or anything like it. So what I

want to talk about is what Mike Dertouzos suggested that nobody would

talk about. I want to talk about the problem of simulating physics with

computers and I mean that in a specific way which I am going to explain.

The reason for doing this is something that I learned about from Ed

Fredkin, and my entire interest in the subject has been inspired by him. It

has to do with learning something about the possibilities of computers, and

also something about possibilities in physics. If we suppose that we know all

the physical laws perfectly, of course we don't have to pay any attention to

computers. It's interesting anyway to entertain oneself with the idea that

we've got something to learn about physical laws; and if I take a relaxed

view here (after all I'm here and not at home) I'll admit that we don't

understand everything.

The first question is, What kind of computer are we going to use to

simulate physics? Computer theory has been developed to a point where it

realizes that it doesn't make any difference; when you get to a

universal

computer,

it doesn't matter how it's manufactured, how it's actually made.

Therefore my question is, Can physics be simulated by a universal com-

puter? I would like to have the elements of this computer

locally intercon-

nected,

and therefore sort of think about cellular automata as an example

(but I don't want to force it). But I do want something involved with the

467

468

Feylaman

locality of interaction. I would not like to think of a very enormous

computer with arbitrary interconnections throughout the entire thing.

Now, what kind of physics are we going to imitate? First, I am going to

describe the possibility of simulating physics in the classical approximation,

a thing which is usuaUy described by local differential equations. But the

physical world is quantum mechanical, and therefore the proper problem is

the simulation of quantum physics--which is what I really want to talk

about, but I'U come to that later. So what kind of simulation do I mean?

There is, of course, a kind of approximate simulation in which you design

numerical algorithms for differential equations, and then use the computer

to compute these algorithms and get an approximate view of what ph2csics

ought to do. That's an interesting subject, but is not what I want to talk

about. I want to talk about the possibility that there is to be an

exact

simulation, that the computer will do

exactly

the same as nature. If this is to

be proved and the type of computer is as I've already explained, then it's

going to be necessary that

everything

that happens in a finite volume of

space and time would have to be exactly analyzable with a finite number of

logical operations. The present theory of physics is not that way, apparently.

It allows space to go down into infinitesimal distances, wavelengths to get

infinitely great, terms to be summed in infinite order, and so forth; and

therefore, if this proposition is right, physical law is wrong.

So good, we already have a suggestion of how we might modify

physical law, and that is the kind of reason why I like to study this sort of

problem. To take an example, we might change the idea that space is

continuous to the idea that space perhaps is a simple lattice and everything

is discrete (so that we can put it into a finite number of digits) and that time

jumps discontinuously. Now let's see what kind of a physical world it would

be or what kind of problem of computation we would have. For example,

the first difficulty that would come out is that the speed of light would

depend slightly on the direction, and there might be other anisotropies in

the physics that we could detect experimentally. They might be very small

anisotropies. Physical knowledge is of course always incomplete, and you

can always say we'll try to design something which beats experiment at the

present time, but which predicts anistropies on some scale to be found later.

That's fine. That would be good physics if you could predict something

consistent with all the known facts and suggest some new fact that we didn't

explain, but I have no specific examples. So I'm not objecting to the fact

that it's anistropic in principle, it's a question of how anistropic. If you tell

me it's so-and-so anistropic, I'll tell you about the experiment with the

lithium atom which shows that the anistropy is less than that much, and

that this here theory of yours is impossible.

Simulating Physics with Computers

469

Another thing that had been suggested early was that natural laws are

reversible, but that computer rules are not. But this turned out to be false;

the computer rules can be reversible, and it has been a very, very useful

thing to notice and to discover that. (Editors' note: see papers by Bennett,

Fredkin, and Toffoli, these Proceedings). This is a place where the relation-

ship of physics and computation has turned itself the other way and told us

something about the possibilities of computation. So this is an interesting

subject because it tells us something about computer rules, and

might

tell us

something about physics.

The rule of simulation that I would like to have is that the number of

computer elements required to simulate a large physical system is only to be

proportional to the space-time volume of the physical system. I don't want

to have an explosion. That is, if you say I want to explain this much physics,

I can do it exactly and I need a certain-sized computer. If doubling the

volume of space and time means I'll need an

exponentially

larger computer,

I consider that against the rules (I make up the rules, I'm allowed to do

that). Let's start with a few interesting questions.

2. SIMULATING TIME

First I'd like to talk about simulating time. We're going to assume it's

discrete. You know that we don't have infinite accuracy in physical mea-

surements so time might be discrete on a scale of less than 10 -27 sec. (You'd

have to have it at least like to this to avoid clashes with experiment--but

make it

10 -41 sec.

if you like, and then you've got us!)

One way in which we simulate timewin cellular automata, for example

--is to say that "the computer goes from state to state." But really, that's

using intuition that involves the idea of time--you're going from state to

state. And therefore the time (by the way, like the space in the case of

cellular automata) is not simulated at all, it's imitated in the computer.

An interesting question comes up: "Is there a way of simulating it,

rather than imitating it?" Well, there's a way of looking at the world that is

called the space-time view, imagining that the points of space and time are

all laid out, so to speak, ahead of time. And then we could say that a

"computer" rule (now computer would be in quotes, because it's not the

standard kind of computer which cperates in time) is: We have a state s~ at

each point i in space-time. (See Figure 1.) The state

s i

at the space time

point i is a given function F,(sj, s k .... ) of the state at the points j, k in some

neighborhood of i:

s,= .... )

470

Feynman

• •

• " "$k

oto

~pac~

Fig. 1.

You'll notice immediately that if this particular function is such that the

value of the function at i only involves the few points behind in time, earlier

than this time i, all I've done is to redescrib6 the cellular automaton,

because it means that you calculate a given point from points at earlier

times, and I can compute the next one and so on, and I can go through this

in that particular order. But just let's us think of a more general kind of

computer, because we might have a more general function. So let's tlaink

about whether we could have a wider case of generality of interconnections

of points in space-time. If F depends on a//the points both in the future and

the past, what then? That could be the way physics works. I'll mention how

our theories go at the moment. It has turned out in many physical theories

that the mathematical equations are quite a bit simplified by imagining such

a thing--by imagining positrons as electrons going backwards in time, and

other things that connect objects forward and backward. The important

question would be, if this computer were laid out, is there in fact an

organized algorithm by which a solution could be laid out, that is, com-

puted? Suppose you know this function F, and it is a function of the

variables in the future as well. How would you lay out numbers so that they

automatically satisfy the above equation? It may not be possible. In the case

of the cellular automaton it is, because from a given row you get the next

row and then the next row, and there's an organized way of doing it. It's an

interesting question whether there are circumstances where you get func-

tions for which you can't think, at least right away, of an organized way of

laying it out. Maybe sort of shake it down from some approximation, or

something, but it's an interesting different type of computation.

Question: "Doesn't this reduce to the ordinary boundary value, as

opposed to initial-value type of calculation?"

Answer: "Yes, but remember this is the computer itself that I'm

describing."

It appears actually that classical physics is causal. You can, in terms of

the information in the past, if you include both momentum and position, or

Simulating Physics with Computers

471

the position at two different times in the past (either way, you need two

pieces of information at each point) calculate the future in principle. So

classical physics is

local, causal, and reversible,

and therefore apparently

quite adaptable (except for the discreteness and so on, which I already

mentioned) to computer simulation. We have no difficulty, in principle,

apparently, with that.

3. SIMULATING PROBABILITY

Turning to quantum mechanics, we know immediately that here we get

only the ability, apparently, to predict probabilities. Might I say im-

mediately, so that you know where I really intend to go, that we always have

had (secret, secret, close.the doors!) we always have had a great deal of

difficulty in understanding the world view that quantum mechanics repre-

sents. At least I do, because I'm an old enough man that I haven't got to the

point that this stuff is obvious to me. Okay, I still get nervous with it. And

therefore, some of the younger students ... you know how it always is,

every new idea, it takes a generation or two until it becomes obvious that

there's no real problem. It has not yet become obvious to me that there's no

real problem. I cannot define the real problem, therefore I suspect there's no

real problem, but I'm note sure there's no real problem. So that's why I like

to investigate things. Can I learn anything from asking this question about

computers--about this may or may not be mystery as to what the world

view of quantum mechanics is? So I know that quantum mechanics seem to

involve probability--and I therefore want to talk about simulating proba-

bility.

Well, one way that we could have a computer that simulates a prob-

abilistic theory, something that has a probability in it, would be to calculate

the probability and then interpret this number to represent nature. For

example, let's suppose that a particle has a probability

P(x, t)

to be at x at a

time t. A typical example of such a probability might satisfy a differential

equation, as, for example, if the particle is diffusing:

al,(x, t) _

v P(x,t)

Now we could discretize t and x and perhaps even the probability itself and

solve this differential equation like we solve any old field equation, and

make an algorithm for it, making it exact by discretization. First there'd be

a problem about discretizing probability. If you are only going to take k

digits it would mean that when the probability is less that 2 -k of something

happening, you say it doesn't happen at all. In practice we do that. If the

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