The Planck length is defined as the square root of the product of the gravitational constant and the reduced Planck’s constant over the speed of light cubed. We can measure all three of these fundamental physical constants, and, so, we can calculate the value of the Planck length, which turns out to be about \(1.616199 \times 10^{-35}\) m. Without the benefit of scientific notation, Planck length is about 0.00000000000000000000000000000000001616199 m. (There is a gigantic margin of error, which is about \(.000097\times 10^{-35}\) m due to the margin of error in measuring the gravitational constant.)Footnote 5

Related to the Planck length is the Planck time, which is the time it takes light in a vacuum to travel one Planck length.Footnote 6 It turns out to be about \(5.391 \times 10^{-44}\) s, or 0.00000000000000000000000000000000000000000005391 s. One can derive the Planck time from the same three fundamental constants (the square root of the product of the gravitational constant and the reduced Planck’s constant over the speed of light to the fifth).Footnote 7

When discussing Planck length or Planck time, one has to be extremely careful to get the physics right. What, then, is their significance? The short answer is: we don’t know. Planck length is an interesting length for one because it is defined entirely in terms of physical constants rather than arbitrary features of aspects of the universe (e.g., arbitrary numbers of cycles of light or the distance light travels in some arbitrary period). Given the significance of the gravitational constant (i.e., the strength of the gravitational force), the reduced Planck constant (i.e., the smallest possible change in angular momentum), the speed of light (i.e., the invariant velocity), and the fact that the Planck length is the natural unit of length that is defined in terms of these constants, it has a natural significance. In other words, if one takes the gravitational constant, the reduced Planck constant, and the speed of light as units of force, angular momentum (action), and velocity, respectively, to define a system of units (often called natural units) then the Planck length is the unit of length and the Planck time is the unit of duration in this system.Footnote 8 In a system of this sort, many natural laws have simpler formulations. Beyond that, there is nothing significant about the Planck length that one can glean from its definition alone.

The Planck length is often said to be a minimal length and the Planck time a minimal duration. However it is rarely made clear what is meant by these claims. Neither of the two major traditions in contemporary physics, Quantum Mechanics (and its development into Quantum Field Theory and the Standard Model of Particle Physics) and Relativity (and its development from Special to General and into the Standard Model of Cosmology), imply anything in particular about the Planck length, nor does either one by itself imply that there is a minimal length. Instead, the definition of the Planck length borrows from each theory. Quantum Mechanics (QM) and General Relativity (GR) are notorious for being incompatible, but the incompatibility is rather subtle and complex. It is not as if one of them states that p and the other states that not p, which would make them just flat inconsistent. Instead, one can use parts of one and parts of the other to get meaningful results (as many of the arguments below will illustrate). However, if one uses the wrong parts, the one gets nonsensical results.Footnote 9

The Planck length has a certain significance that derives from QM and GR together—it is significant because one can argue from certain aspects of QM and GR that there is a minimal length (in a sense to be clarified below) and that it is near the Planck length (by ‘near’ I mean within a couple of orders of magnitude). Moreover, the Planck length is the scale below around which one needs a successor to both QM and GR to figure out what happens (i.e., the nonsensical answers one sometimes gets from combining QM and GR occur for phenomena at the Planck length and smaller). There are many theories that are designed to incorporate some aspects of QM and some aspects of GR without causing these problems. It is common to call these theories of quantum gravity (although that can be misleading because ‘quantum gravity’ sometimes refers to specific ones of these theories). They include loop quantum gravity, superstring theory, noncommutative geometry, causal sets, and doubly special relativity (I discuss some of these below).Footnote 10 Many of these successor theories imply that there is a minimal length (often in very strong ways) and many also imply that it is near the Planck length. So, the significance of the Planck length is subtle and depends on whether one is considering just its definition, some aspects of QM and GR, or quantum gravity.Footnote 11 As such it can be misleading to say that a theory of quantum gravity entails that the Planck length is a quantum of spacetime—it is better to say that the theory entails that there is a minimal length and that it is near the Planck length. For this reason and the fact that a more general presentation will be beneficial later, I focus on whether there is a minimal length, regardless of whether it happens to be identical to the Planck length. I designate the minimal length \(\ell _{\mathrm{M}}\).

There are several things one might mean by saying that \(\ell _{\mathrm{M}}\) is the minimal length or distance.Footnote 12 One might mean that:

(1) spacetime is quantized or granular or discrete (one sees all these terms in the literature), which entails that spatial quantities come in chunks and can only have values that are multiples of \(\ell _{\mathrm{M}}\). (2) ‘distance’ has no meaning for values less than \(\ell _{\mathrm{M}}\).Footnote 13 (3) there is no such thing as a distance shorter than \(\ell _{\mathrm{M}}\). (4) it is impossible to measure a distance less than \(\ell _{\mathrm{M}}\). (5) any scale to measure length must have a minimal value of \(\ell _{\mathrm{M}}\) (i.e., have a unit that is an integral multiple of \(\ell _{\mathrm{M}})\).Footnote 14

Some discussion of these minimal length theses is in order.

(1) Spacetime is quantized This is a claim exclusively about the physical world—it says nothing at all about how we think about the physical world or how we represent the physical world or about the language we use to talk about the physical world. It also says nothing at all about our knowledge of the physical world. In essence, it says that spacetime has a particular structure, namely, that there is no such thing as a distance that is not an integral multiple of the minimal distance. We might formulate this point in terms of existence—the only lengths or distances that exist are those that are integral multiples of the minimal length. There exist no other distances.

We already know (or at least we have a colossal amount of evidence to support) that certain things are quantized, like the energy of an atom and the frequency of light. The claim about spacetime being quantized is meant to be similar to these familiar claims. The claim that energy is quantized comes down to the claim that it is impossible for there to be certain energy states; as something changes energy, the increase or decrease in energy occurs in “packets” or quanta. It is impossible for there to be an energy state in between the units of the quanta. Likewise, the claim that spacetime is quantized comes down to the claim that it is impossible for there to be certain distances or durations; as the spatial distance between two things changes or the temporal interval between two things changes, the increase or decrease in distance or duration occurs in “packets” or quanta. It is impossible for there to be a distance or duration in between the units of the quanta.

(2) ‘Distance’ has no meaning for values less than the minimal length This is a claim about the meaning of a particular English word, ‘distance’. As such, it is not a claim about the physical world, but rather a claim about our language. It might seem like an odd way to capture the claim that there is a minimal length, but I recommend thinking of it in the following way. Intuitively, it might seem that any positive number might be a real distance, but it turns out that there are certain values (any number less than the minimal length in the given units) the cannot be real distances. In other words, beyond the minimal length, there is nothing to refer to when using the word ‘distance’. Therefore, I think this sort of statement is just a roundabout way of saying that there are no distances below a certain minimal value.

One difference between claims (1) and (2) is that the latter is entirely about spatial intervals (distances) and not about temporal intervals (durations). However, special and general relativity entail that spatial and temporal intervals cannot be neatly separated. Therefore, I assume throughout that what goes for spatial intervals goes for temporal intervals as well.

(3) There is no such thing as a distance shorter than the minimal length This is a claim about the physical world, not a claim about our language or thought or knowledge. It is essentially the same claim as the one made by (2), when (2) is properly interpreted as being about the world rather than about the meaning of ‘distance’. Both say that distances less than the minimal length do not exist. That is, there simply are no spatial intervals less than the minimal length. Again, what goes for distance goes for duration as well; so, given our background information, claim (3) should be interpreted as entailing that there is no such thing as a duration less than the minimal duration.

(4) It is impossible to measure a distance less than the minimal length This is a claim about what it is possible to measure, and so it should be interpreted as a claim about what in principle can be done in human practice. In particular, it should be interpreted as a limit to what anyone can measure in practice. As such, it is a practical claim about what humans (or maybe any rational entity) can do. In this presentation, I am assuming that ‘measure’ is a success term; if someone measures a length of certain sort, then there is a length of that sort. If the reader does not share this assumption, then such a reader should replace all occurrences of ‘measure’ with ‘successfully measure’. It should be clear that if there is no such thing as a distance less than the minimal length, then it is impossible to measure a distance less than the minimal length. However, the converse does not hold—it might be that there are certain lengths despite our (in principle) inability to measure them. If (4) turned out to be true, then it would be analogous to an epistemic reading of the uncertainty principle in quantum mechanics (for example, in hidden variable theories), which states that the certain properties cannot be measured despite there being a fact of the matter about their values. There are, of course, myriad controversies and difficulties with interpreting quantum mechanics, so this analogy should be taken as a heuristic, which is intended as an aid to understanding.

(5) Any scale to measure length must have a minimal value of the minimal length This claim is not about the world nor is it about how we talk or think, nor is it about what we can know about the world. Instead, it is a claim about any legitimate measurement scale for length. It has been said already, but it is worth emphasizing that the International System of Units (SI) is far and away the most common and popular in the world, and it employs the meter as its basic unit of length. If it turns out that the meter is not an integral multiple of the minimal length, which, as we will see, is very likely, then any measurement system that includes the meter would be unacceptable. If there is a minimal length, and the meter turns out to not be an integral multiple of minimal lengths, then why would that be a problem? The problem is that a measurement system like this would not accurately represent reality. There would be no such thing as the distance defined as a meter. There would be distances a bit shorter and there would be distances a bit longer, but there would be no such thing as a distance that is exactly a meter. Consider an analogy with some quantity we already think is quantized, like energy. Any system of measurement that includes a unit for measuring energy that is not an integral multiple of the minimal energy is obviously problematic. The same goes for measuring distance in the event that there is a minimal length.

We need to establish the logical relationships between these five interpretations. Based on the discussion above, it should be clear that (1) entails all the rest. Moreover, when properly interpreted, (1), (2), and (3) are equivalent. That is, spacetime is quantized iff there is nothing smaller than the minimal distance for ‘distance’ to refer to (and nothing shorter than the minimal duration for ‘duration’ to refer to) iff there is no such thing as a distance smaller than the minimal distance (and no such thing as a duration shorter than the minimal duration).

It should be obvious as well that (1) entails (5). One might think that (2) and (3) do not entail (5); for example, one think that there could be no distance values less than 5 but anything greater is fair game. But then if entity A is 5 distant from entity B and entity C is 6 distant from B in the same direction, then it looks like A is 1 distant from C, which violates our assumption. Thus, it looks like (2) and (3) entail (5) (given some basic assumptions about the acceptability of a measurement scale). It should be equally clear that (5) does not entail (1), (2) or (3). After all, it might be that considerations pertaining to the nature of measurement scales mandate (5) but these do not entail anything about whether there exist distances less than the minimal length (or durations less than the minimal duration).

(4) does not entail (5); i.e., (4) is compatible with the scale of measurements greater than \(\ell _{\mathrm{M}}\) being continuous (i.e., it is not the case any value of a metric—distance function—is an integral multiple of \(\ell _{\mathrm{M}})\). Why doesn’t the same problem as above occur with (4) as with (2) and (3)? The reason is that there is a distinction between measurement and calculation. Say one measures the distance between event A and event B as 1.5\(\ell _{\mathrm{M}}\) and between event A and event C in the same direction as 2\(\ell _{\mathrm{M}}\). Neither measurement violates (4). We then calculate the distance between event B and event C as .5\(\ell _{\mathrm{M}}\). The calculation does not violate (4) either.

Overall, (1), (2), and (3)—properly interpreted—are equivalent. They entail (4) and (5). Our focus throughout what follows will be (5) and it will turn out that we will not be concerned with (4) after this section.

Why believe that there is a minimal length? There are plenty of arguments for minimal length theses, but they are rarely if ever distinguished by which minimal length thesis is the conclusion of the argument. Let us review some of them.Footnote 15

A. Light and distance When measuring a distance by measuring the time it takes light to traverse it, the accuracy of the measurement increases as the wavelength of the light decreases. However, as the wavelength of the light decreases, its energy increases. As the energy increases, it deforms spacetime more. An analysis of these relationships reveals that once one decreases the wavelength of the light past a certain point, the spacetime deformation decreases the accuracy of the measurement, so there is a limit to how accurate such a measurement can be. It turns out to be around a Planck length. This argument supports interpretation (4). B. Light and volume If we try to measure some properties of a region of space then we need to use light with high enough energy so that the region does not change while we measure it. Once one increases the energy enough, the light deforms the spacetime of the region so much that it no longer constitutes an accurate measurement. This limit occurs at about the Planck length. This argument supports interpretation (4). C. Density If we begin with some mass in a certain regular volume and begin increasing its density by decreasing the volume, then, according to GR, we eventually reach a point at which the process stops because we create a black hole; the radius of the volume at which this occurs is proportional to the mass (i.e., smaller masses result in a smaller radius) According to QM, the same system eventually reaches a point at which the process stops because the uncertainty of the energy in the system reaches a maximum; the radius of the volume at which this occurs is inversely proportional to the mass (i.e., larger masses result in a smaller radius). Using these two results, we can solve for the smallest radius possible, which turns out to be about the Planck length. Because this argument is not about measurement, it seems to support an interpretation stronger than (4), like (2) or (3). D. Uncertainty and position The uncertainty principle from QM entails that there is an inverse relationship between the precision with which we may measure a particle’s momentum and its position. If we alter this principle to include the effect of gravity, then we arrive at the generalized uncertainty principle, which entails that any particle has a minimum position uncertainty of around a Planck length, no matter how uncertain its momentum is. If position measurements have a minimum uncertainty of a Planck length, then it is impossible to measure any distance less than a Planck length, so this argument supports interpretation (4). However, if the uncertainty in question is taken to be with respect to position itself rather than our measurement of position, then this argument supports a stronger interpretation like (2) or (3). E. Energy density and gravity A gravitational field has a certain energy and the density of that energy is related to the strength of the field. Because of the energy-time uncertainty in QM, any region has some fluctuations in gravity that limit our measurements of gravitational energy in that region. These fluctuations in energy correspond to distortions in the spacetime of the region. It turns out that uncertainty in the energy density of the field corresponds to an uncertainty in the spatial specifications of the region. The specifications of the region are defined only to about the Planck length. This argument supports interpretation (4) for sure, and if the uncertainty in question pertains to energy itself rather than our measurements of it, then the argument seems to support stronger interpretations as well (e.g., (2) and (3)). F. Weakness of gravity Of the four acknowledged fundamental forces (i.e., gravity, electromagnetism, the weak nuclear force, and the strong nuclear force), gravity is the weakest, and it is dramatically weaker than the others. The standard model of particle physics incorporates theories of electromagnetism, the weak force, and the strong force. However, at around the order of the Planck length, gravitational effects of particles described by the standard model are no longer negligible. That is, one needs to account for gravitational effects when considering processes that occur around the scale of the Planck length. That is exactly what we cannot do in a straightforward way because of problems integrating GR and QM. Thus, it is natural to think that we need some new physical theory to describe processes that occur around the Planck length. This argument does not directly support any of the interpretations because of its heuristic character, but if we think of distance as implicitly defined by our best physical theories (e.g., QM and GR), then the concept of distance no longer makes sense at scales less than the Planck length. This consideration supports interpretation (2). G. Superstring theory An attempt to unify QM and GR is superstring theory, which posits very small strings whose features explain the central claims of QM and GR. These strings are around the size of the Planck length. It is unclear which interpretation of minimal length superstring theory supports, but it is reasonable to think that it is (2) or (1). H. Loop quantum gravity An alternative attempt to unify QM and GR is loop quantum gravity, which describes spacetime geometry in a novel way so that area and volume are quantized. Loop quantum gravity supports interpretation (1) of minimal length.

Arguments (A)-(F) appeal to aspects of QM and GR, while arguments (G) and (H) depend on theories that are designed to incorporate both QM and GR while avoiding the problems we have applying them together. It is these successor theories that provide us with the arguments for the strongest versions of the minimal length thesis.

In what follows, it is the claim that any meaningful distance must be an integral multiple of the minimal length that plays a central role. This claim is supported by theses (1), (2), (3), and (5), and we have strong arguments for these theses. Thus, we have good reason to believe that there is a minimal length in a strong sense even if we ignore thesis (4) and the arguments for it.

On the other hand, there is some evidence against the Planck length being a minimal length. This evidence usually falls into one of two types: (i) theoretical calculations of meaningful distances less than the Planck length and (ii) astrophysical measurements of the smoothness of spacetime.Footnote 16 For example, in the famous paper by Bekenstein, he calculates that if one adds one photon to a solar mass black hole, then its radius would increase by about 10\(^{-71 }\) m. The astrophysical measurements typically use theoretical considerations to predict that a minimal length of spacetime would cause certain features (e.g., polarization) in light that travels through spacetime, and over long enough intervals, we would be able to detect these features. So far, we have not detected any.

I do not think that the calculations constitute evidence against there being a minimal length because these calculations are usually based on GR, which is consistent with there being no minimal length. So it is not a surprise to find out that one can use GR to calculate something being smaller than the Planck length. Figure 1 shows which combinations of masses and distances are ruled out by QM and GR.Footnote 17 In it, the vertical axis is length in centimeters and the horizontal axis is mass in grams. One arrives at a minimal length only by considering both QM and GR (and the right aspects of them at that).

The astrophysical measurements on the other hand do provide evidence against some values of a minimal length, but they are far from conclusive. For example, they often rely on Lorentz invariance considerations (i.e., a minimal length would be invariant across reference frames), but this is a controversial assumption.Footnote 18 Moreover, at best, they tell against certain values of a minimal length, not against a minimal length per se.

To sum up, we do not know whether there is a minimal length in any of the above senses. We do not know whether a minimal length would be the Planck length or some other length. We do have many good reasons to think that there is a minimal length. Think of what follows as an investigation into the consequences for the SI system if there is a minimal length in the sense of (5), which is entailed by (1), (2), and (3).