# Why is it more convenient to consider spacetime as a continuum?

I often find that phisicists and cosmologists make use of Planck's units. I have read propositions that sound like

"...at the level of Planck's units many law of physics break down"

"...Planck time, the smallest observable unit of time...before which science is unable do describe the universe"

"... it would become impossible to determine the difference between two locations less than one Planck length apart"

even in string theory:

"* Planck length is the order of magnitude of the oscillating strings that form elementary particles, and shorter lengths do not make physical sense*".

Notwithstanding this and the fact that in QM (almost) everything is quantizied (discrete) I read that in mainstream they are still considered not discrete, cf. Phys.SE link.

I am not particularly aware of any pros, I see only cons; can someone tell me what are the compelling reasons to consider spacetime not discrete? Is it a requisite of relativity? As a corollary, I suppose that they must be both discrete or non discrete, right?

• Even if the univers is discrete in time in space (I like to believe that), it tries real hard to conceal that. For example, ultra-high-frequency light is not slowed down by the coarseness of space, or at least is slowed down much less than one would expect. – John Dvorak Aug 9 '14 at 8:50
• The primary benefit of regarding space as a continuum is calculus. – David H Aug 9 '14 at 9:00
• I have the impression that discrete spacetime will not work with special relativity, which has been tested and validated at the limits of possible measurements. physics.stackexchange.com/q/33273 – anna v Aug 9 '14 at 9:17
• @bobie Because plancke units are too large. Suppose you have a circle with radius equal to one plancke unit. If you try to calculate the area of the circle of this unit circle via "integration" with $dA$ equal one plancke unit-squared, you get an area of 4 units-squared instead of $\pi$. – David H Aug 9 '14 at 10:43
• Strongly related, if not a duplicate: physics.stackexchange.com/q/9720 – Kyle Kanos Jan 28 '15 at 13:58

The main reason why physics isn't building on the assumption that the time is discrete is the fact that such an assumption is demonstrably incorrect. Physics is a natural science, a process of learning how Nature actually does work, not a movement to irrationally and indefensibly claim that there are some "cons" or "pros" about some arbitrary philosophical positions how Nature should work.

Time has to be described as a continuous variable because the Lorentz symmetry, the symmetry underlying relativity, a pillar of physics, is a symmetry continuously transforming continuous time and continuous space.

Moreover, all the evolution equations – equations of motion in classical mechanics, field equations in classical field theory, and/or Schrödinger's, Heisenberg's, or other equations governing any quantum mechanical theory – are differential equations for functions of time that, as David H said, couldn't work if time failed to be continuous. In quantum mechanics, one would really have to sacrifice any agreement for the experiment (by making the time really discrete) or to sacrifice unitarity because all generic enough quasi-continuous but not continuous transformations of the Hilbert space would fail to be unitary.

So the Planck time is the minimum duration beneath which time surely behaves differently and the everyday life statements about the time break down or cease to hold. But what replaces them is certainly not a naive picture of a discrete time that is counted by integers like apples.

• *...such an assumption is demonstrably incorrect.*Can you provide proof or link? – bobie Aug 10 '14 at 4:35
• Dear Bobie, yes, I can, but the proof is included in this very answer, too. It is really a trivial thing. Competent physicists aren't really writing many papers about this pet idea of some laymen ("discrete time") because the right answer is so obvious. – Luboš Motl Aug 10 '14 at 7:55
• Please highlight the passage of your answer that you consider a proof. – bobie Aug 10 '14 at 8:21
• I think the wording of your first sentence is causing problems. You say "demonstrably incorrect". But your subsequent paragraphs say that it's inconsistent with the math models we use. That doesn't mean it can't be possibly true. – garyp Apr 11 '20 at 18:32
• It does because the validity of the relevant traits of these mathematical models is a demonstrable fact. It is your math phobia and not my answer that is actually causing your problems. – Luboš Motl Apr 12 '20 at 19:05

The question is not whether spacetime is discrete, but whether it is substantive. If you assume a substantive spacetime, then it cannot be discrete because that would violate fundamental symmetry principles. It would violate relativity as described in the Lorentz transform.

If, on the other hand, you consider spacetime as consisting only of the results (potential and actual) of physical measurement, and having no prior existence of its own, then it is naturally discrete at the level of the resolution of measurement. This does not violate relativity, because measurement is relative to the reference frame (or reference matter) chosen by the observer.

It is worth noting that in the axiomatic description of quantum mechanics due to von Neumann and Dirac, and quantum mechanics is a theory of measurement results. In in quantum logic (also due to von Neumann) it is actually the formal mathematical structure of a language for describing relationships in measurement results. From this point of view, there is no reason to assume that spacetime is substantive. All of known physics can be described without such an assumption. Showing that this is true, both conceptually and mathematically has been a central theme of my books.

The Large and the Small

The Mathematics of Gravity and Quanta