# 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
• It's the 2nd paragraph and the first half of the 3rd paragraph. – Luboš Motl Aug 10 '14 at 16:22