# Is time continuous or discrete?

I was coding a physics simulation, and noticed that I was using discrete time. That is, there was an update mechanism advancing the simulation for a fixed amount of time repeatedly, emulating a changing system.

I though that was interesting, and now believe the real world must behave just like my program does. It is actually advancing forward in tiny but discrete time intervals?

## 13 Answers

As we cannot resolve arbitrarily small time intervals, what is ''really'' the case cannot be decided.

But in classical and quantum mechanics (i.e., in most of physics), time is treated as continuous.

Physics would become very awkward if expressed in terms of a discrete time: The discrete case is essentially untractable since analysis (the tool created by Newton, in a sense the father of modern physics) can no longer be applied.

Edit: If time appears discrete (or continuous) at some level, it could still be continuous (or discrete) at higher resolution. This is due to general reasons that have nothing to do with time per se. I explain it by analogy: For example, line spectra look discrete, but upon higher resolution one sees that they have a line width with a physical meaning.

Thus one cannot definitely resolve the question with finitely many observations of finite accuracy, no matter how contrived the experiment.

• Unless we resolved it by determining that time is, in fact, discrete. – aquirdturtle Oct 16 '18 at 6:36
• the tool created by Newton - well, most of modern integration notation and usage actually stems from Leibniz and thus differential equations are significantly easier to use, it might be worth an honourable mention – Narusan Oct 16 '18 at 10:55
• @Narusan: I didn't intend to give a full history. Newton, not Leibniz, introduced calculus as a tool for studying physics.The dot notation of Newton for time derivatives is still used in modern physics. – Arnold Neumaier Oct 16 '18 at 11:10
• While this is right, on the other hand, the name calculus stems from Leibniz (no-one calls it flux), so does the integration sign, second and higher derivatives and so on are only possible because of Leibniz' discoveries. I'm not saying he was the true creator of calculus, but that both deserve due credit as both developed calculus independently of each other. Therefore, to call calculus a "the tool created by Newton" seems oversimplified, you could a) just leave it out as it doesn't matter to the post, or b) give proper credit if you chose to give credit. – Narusan Oct 16 '18 at 15:42
• @Alchimista: In qunatum mechanics and in quantum field theory, time is a continuous parameter not subject to quantization. The time-energy uncertainty relation is quite different in derivation and meaning from the position-momentum uncertainty relation, see physics.stackexchange.com/questions/365873 – Arnold Neumaier Jan 7 at 9:04

I think it's important to note that quantum or quantized time is not equal to discrete time. For instance, we have "quantized" space. By this we mean that it receives quantum treatment. But the underlying coordinates still form a continuum. So even if you live on a finite circle and only consider wavefunctions so that you get a countable set of basis functions from which to form all the others, you can still in principle measure incidence of particles at any point, again forming a continuum. Therefore, if we take quantum time in analogy to quantum space, we would have to conclude that quantum mechanically it would still form a continuum.

Of course none of this proves how the universe really works, which is your question. The only honest answer direct to your question is "We don't know". Physical theories do not describe how the universe actually works, the only thing we know is that their predictions match experimental results we currently posses. So even if the best physical theories we currently posses use a continuum of temporal coordinates, we cannot by any means conclude that the way the universe actually works matches our description.

• We have quantized space? News to me. We do have quantized angular momentum and other variables, which behaves the way you describe quantized variables as working. – Peter Shor Jun 18 '13 at 22:28
• I just mean that there exist quantum observables corresponding to position, and their outcomes in general form a continuum. Time is another issue. – SMeznaric Jun 19 '13 at 20:38
• @Peter Shor: I recall reading that LQG has area and volume operators with discrete spectrum. – Mozibur Ullah Nov 4 '17 at 21:43
• @Mozibur Ullah: I agree that lots of physicists think space is quantized. I was taking issue with this answer asserting it as an absolute fact. – Peter Shor Nov 4 '17 at 22:06

The answer to this question is not known presently. Current physics is, as stated by other answers, based on fully continuous mathematical models, which particularly assume spacetime to be continuous. On the other hand you could argue that these models are isomorphic to discrete constructive models, with the general view that the continuous is the limit of the discrete. Some modern spacetime theories assume an underlying network/relational structure, and are fully discrete.

My personal belief is that continuous structures do not exist in the physical world. This is however just a belief.

See also: Is the universe finite and discrete?

What you are talking about is similar to the problem of quantum gravity. Since gravity is an effect of the curvature of spacetime, to have a quantum theory of it, you need to quantize the spacetime manifold. This is done with spin foams which are little units of volume in spacetime that have spins associated to them. They connect together like total angular momentum and build up into various kinds of geometry. This is just a theory, but comes from the very real problem of "what is the quantum field theory of gravity". Also, it answers the question "Higher power is needed to resolve smaller dimensions (sizes). To resolve small enough distances, the power eventually gets large enough to couple to the metric of space time. How do we talk about spacetime when the uncertainty in the injected energy transfers to uncertainty in the metric."

I'd say there's no conclusive evidence, but in quantum physics, Planck time is sometimes cited as a possible smallest unit of time.

The source for my data is Quantum Gods: Creation, Chaos, and the Search for Cosmic Consciousness by Victor J. Stenger. In there, he goes into a lot of detail about this in one chapter.

• From Wikipedia: Within the framework of the laws of physics as we understand them today, for times less than one Planck time apart, we can neither measure nor detect any change. So it's not necessarily the smallest unit of time, just the smallest one we're capable of using. – Brendan Long Sep 5 '12 at 14:34
• @BrendanLong - Except there's the philosophical question of "If there's no way to measure it, does it even exist?". Largely, for example, the answer for Heisenberg's uncertainty principle is that the information about a particles position and velocity don't actually physically exist simultaneously. So, if we can't measure a unit of time smaller than Planck time, if it's physically impossible, then perhaps it doesn't even exist. – Omnifarious Sep 5 '12 at 15:17
• My interpretation of the Planck Time is that it's the smallest meaningful unit of time. Time itself is continuous, i.e. intervals shorter than the Planck Time exist. But these shorter intervals are trivial, so time may as well be discrete. Additionally, if time is discrete then distance as well must be discrete. It's weird to think of the universe as pixelated... – chharvey Sep 5 '12 at 23:25
• -1. Quantum Mechanics regards spacetime as continuous, and that includes time too!. – Abhimanyu Pallavi Sudhir Jun 22 '13 at 14:16
• -1. This is a classic pop science argument. The logic seems to be "the Planck time is a time... so therefore every time is a multiple of it". (Note that this same argument makes as much sense, i.e. none at all, for literally any other quantity.) See discussion here. – knzhou Jan 28 '18 at 17:01

Due to the work of Julian Barbour and others, time is defined (in a closed system) by keeping track of all the changes (of particles and so on).

In this respect we would say that in a classical system (macroscopic) that time would be continuous since the motions of such objects are essentially continuous and the way that you parameterize the changes would then be continuous.

In a quantum mechanical system, i think this gets trickier because the formalism is kind of set up from the POV of a "scientist in a lab" so that time is continuous classical external parameter for the macroscopic scientist.

In some formulations of QM, position is a continuous variable and particles have definite (but uncertain) position, in this context you can still have a continuous time parameter.

There is no continuous time or space. Only events are happening. Suppose if you are reading this answer is an event. And then looking on the roof is another event. So combine these two based on the measure of time elapse,will get the actual motion of events. same as that in the movies.

• To those downvoting this, I'd like to point out that it is not completely without merit. The work 'Science without Numbers' and the resulting research efforts have successfully formulated various fields of physics without reference to any mathematical objects (numbers, functions, sets, categories, calculus etc.) and, with relevance to this post, without coordinatising space. See also Tarski's axioms (euclidean geometry without sets) and these notes goo.gl/vxYtOA from a lecture by Prof Frank Arntzenius. – ComptonScattering Sep 11 '13 at 23:17
• Was this also suggested by you yourself ? . – Abhimanyu Pallavi Sudhir Sep 13 '13 at 1:50
• @ComptonScattering: But this is non - mainstream . – Abhimanyu Pallavi Sudhir Sep 13 '13 at 1:51
• The edit was not proposed by me. Also I disagree that geometric constructions of science are in any sense fringe if that is what you are suggesting. They are the accepted works of respected scientists. It was merely an exercise to show that algebra, though useful, is not fundamental to what science does, and so one should not promote algebraic to the status of existential when attempting to interpret a theory. In other words, something that was invoked to do a calculation cannot reasonably be said to therefore exist. – ComptonScattering Sep 13 '13 at 8:14
• @ComptonScattering: I was asking wilfred, not you . – Abhimanyu Pallavi Sudhir Sep 13 '13 at 12:15

My understanding of the fundamental issue of Time, is that if we base it upon physical transactions, then we are dealing with a discretized system (e.g. quantized interactions).

Not only that, moreover, a discretized / quantized Time may then have geometric properties that further confound the question.

Many things in physics are quantized (eg: angular momentum of objects, mass of objects, momentum of a particle in a box) so why not time? Well, it maybe in some future theory, but right now what works to explain nature is the idea that the transformations we do to obects behave like Lie Group transformations (eg: rotations, boosts, spacetial translations, time translation, strains). These transformations are labeled by continuous parameters (eg: rotation angles, velocity, boost parameters, translation distance, waiting time, radians of strain). For a Lie Group the parameters are continuous numbers and the group generators (eg: $\vec {J}, \vec {K}, \vec {P}, E$) have quantized eigenvalues. This Lie Group concept divides the quantities in physics into continuous numbers and quantized generators. Time is continuous just like rotation angles are continous.

• The translation generators of the group of Euclidean motions don't have quantized eigenvalues! Neither have those of the Poincare group. – Arnold Neumaier May 2 '18 at 11:13
• @Arnold Yes, I generalized too much. We presently use abelian translation generators in many places in physics. However, the Poincare group is not the end of the story. Since all masses are quantized, I have assumed this will be explained eventually by the non-commutation of translations just as the quantization of angular momentum is explained by the non-commutation of rotations. Even for abelian translations I would still argue that time is continuous because it is a Lie group parameter just like a rotation angle. – Gary Godfrey May 2 '18 at 16:03
• The unitary representations of SL(2) (and hence all groups containing an isomorphic copy of it, in particular all noncompact semisimple Lie groups) also have generators with continuous spectrum. Thus this property has nothing to do with being abelian. Of course this doesn't change the fact that , to have a tractable theory, time must be treated as continuous on a fundamental level. – Arnold Neumaier May 2 '18 at 16:09
• @Arnold More interesting for physics are the finite dimensional irreducible reps of SU(2). The eigenvalues of its generators explain the quantized angular momenta of particles. If the rotation generators all commuted, there would be no raising/lowering operators to zero the m=-j and m=+j kets for integer and half-integer j. That is, non-abelian rotation generators are necessary to quantize angular momentum. Perhaps there will be a similar story that non-abelian translation generators are necessary to quantize mass. – Gary Godfrey May 2 '18 at 21:15

I would like to offer a gedanken approach to that question.

Imagine you are at a stationary system (the lab frame) $S$ in which time is discrete, with regard to some smallest increment of time ($Δt=κ$, a "chronon" if you will).

Within that frame, the maximal acceleration possible, $α$, is given by the maximal change in velocity (from going at the speed of light $c$ in the positive direction to going at $c$ in the negative direction- $Δv=2c$) over the minimal duration in which it can happen (a single chronon- $Δt=κ$). We get:

$$α= \frac{2c}{κ}$$

(In a similiar manner maximal jerk, pop etc. can be derived)

I am not quite sure how I should regard $α$. A maximal acceleration with respect to a frame sounds a bit off to me. If it is identical in all frames we recieve something like one of the the postulates of special relativity- which might sprout higher orders of special relativity. If it is not identical for all frames, then I am not even sure how it can be treated. Note that this is not a maximal acceleration in the object's own rest frame (like the Schwinger limit), but a frame-dependant limit.

Another approach I can think of is that in a discrete-time world, time translation symmetry becomes discrete time translation symmetry, which means conservation of energy is replaced by some discrete time parallel, which I am not familiar with. Maybe nonconsevations of energy can be used to suggest for or against the idea of discrete time.

If anyone could help me continue one of the two lines of thought introduced here, I think discrete time can lead to a contradiction, but I can't quite get there myself.

• How did you measure the velocity change? Wouldn't you need two time steps for that? – Emil Feb 20 '18 at 13:24
• You might be right, I am not sure if the notion of changing velocity over the duration of a single chronon is meaningful (but if it isn't, how can velocity even change at all?). However this does not change the argument, that a maximal acceleration must exist in such a universe... – A. Ok Feb 20 '18 at 15:16
• I can visualize a maximum displacement to not go above speed of light. But acceleration is related to curvature isn't it? Perhaps space can contract or expand continuously but be built up with legos/bins? – Emil Feb 20 '18 at 17:14
• Do the bins conract or expend continuously themselves? Because then I find it hard to see what a change in "the size" of the fundumental quantum of space-time means, as all size is defined by this quantum of space-time... – A. Ok Feb 20 '18 at 17:27
• Sorry, I mixed this question up with another that quantized space aswell. My thought was that the distances between the bins could change but not the distances inside the bins. – Emil Feb 20 '18 at 18:59

To answer your question, time may be advancing forward in tiny but discrete time intervals. If your model reflects or predicts the realty then it is at least just as good as any other. The only awkward part might be that you discretize the continuous/differentiable theory in order to create your simulations. Then the latter might seem superior. It would be nice instead to have an independent theory as a foundation for what you do. I suggest discrete calculus as a starting point. Its idea is simple: $$\lim_{\Delta x\to 0}\left( \begin{array}{cc}\text{ discrete }\\ \text{ calculus }\end{array} \right)= \text{ calculus }.$$

By the very fact of calling it time (i.e. assuming division), infinity appears as discrete. Otherwise, it is continuous. Same for space, since it's the other side of the same coin.

• Please go into more detail about your assumptions and how they leads to your conclusions. As it stands, this doesn't really answer the question. – Benjamin Hodgson Oct 17 '12 at 22:20

Obviously space is continuous, so is time. What is not continuous is the conception of numbers which we are using in computers and measurments. There is no reason for something so fundamental to be descrete, since continuity is more general (and amazing) than discreteness.

• Hmm, I'd need some argumentation. And personally, I like discreteness more than continuity. – jcora Sep 6 '12 at 20:14
• Is your thinking process continuous? If so, so is universe and its time. There is always something inbetween descrete stuff, this emptiness makes whole system continuous. Descreteness is always embedded in continuity, since it needs a separator. – Asphir Dom Sep 6 '12 at 20:27
• That's quite contradictory. Also, you cannot objectively judge your thinking process. It might appear continuous, although neural activity does have steps, but that's not the proof that it is continuous. – jcora Sep 6 '12 at 21:47
• I would argue the opposite, that it is possible to have something discrete without anything in between. For instance, consider a hypothetical sequence of causal events that are not separated by time. The ordering is preserved, but there is literally nothing in between each event. – Michael Oct 16 '18 at 17:19

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