The Quantum Structure of Space and Time (293 pages)

204 The Quantum Structure of Space and Time

5.4 Discussion

G. ’t Hooft I would like to make sort of a claim or statement and then a question.
Actually it bears on Nati’s talk, but also others have mentioned emergent space

and emergent time, and I claim that any theory you have allows a rigorous

definition of time, not a fuzzy one, and even a rigorous definition of space, and

not a fuzzy definition of space. And the argument goes as follows. Assume

you have some theory that is supposed to explain some phenomenon. A priori

there was no space, no time in the question you’ve been asking. You just

have a theory. Then the theory will contain variables and equations, and a

lot of prescriptions how to solve these equations, if it is a good theory. And I

claim that, as soon as you have indicated the order by which you have to solve

the equations, that order defines causality in your theory, and that defines a

notion of time. So time is basically the order by which you have to solve the

equations. If you think a little bit, that’s exactly for instance how a theory of

the planetary system works. The time, the notion of time among the planets

is the order by which you solve the equations. If you solve the equations in

the wrong order, you might have forgotten that two planets might collide, and

then you get impossible answers. So you have to know, exactly that time is the

order by which you solve the equations. And, so that is a rigorous definition,

there’s no way to fool around with that, because if you solve them in the wrong

order, you might get the wrong answer. Similarly however, you can also make

a rigorous definition of space. And that is because, well, I must assume some

form of reduction of a theory into simple equations. If you write down infinitely

complicated equations, you don’t really know what you’re doing, you have to

reduce them to simple equations. And then you can ask, two sets of variables,

how many equations are they away from each other? And that defines a distance

between variables, and that eventually defines space. If you think a little bit,

that’s the way our present space-time seems to work, that two systems are far

away if you have to solve differential equations very very many times before

you reach from one point to the other point. So I think that any theory should

contain some notion of space as well as time, and in a discussion during the

break, the question was asked: does this defines a continuous space-time? And

I would say no, most time and space would be discrete in this sense, but two

variables like that, connected with by an equation, that defines them to be

nearest neighbours, that defines a distance one. And then, so any theory in

some sense looks like a lattice.

J. Harvey I cannot resist comparing, that if this defines not only the order of

time, but the rate at which time proceeds, then I am sure time proceeds much

faster in your part of the world than in my part of the world.

G. ’t Hooft But anyway, the question comes then, that if you would drop the

notion of reduction, then what are you doing, and is that not a direct contra-