Tag Archives: Quantum gravity

A meditation on physical units: Part 1

[Preface: A while back, Michael Raymer, a professor at the University of Oregon, drew my attention to a curious paper by Craig Holt, who tragically passed away in 2014 [1]. Michael wrote:
“Dear Jacques … I would be very interested in knowing your opinion of this paper,
since Craig was not a professional academic, and had little community in
which to promote the ideas. He was one of the most brilliant PhD students
in my graduate classes back in the 1970s, turned down an opportunity to
interview for a position with John Wheeler, worked in industry until age
50 when he retired in order to spend the rest of his time in self study.
In his paper he takes a Machian view, emphasizing the relational nature of
all physical quantities even in classical physics. I can’t vouch for the
technical correctness of all of his results, but I am sure they are
inspiring.”

The paper makes for an interesting read because Holt, unencumbered by contemporary fashions, freely questions some standard assumptions about the meaning of `mass’ in physics. Probably because it was a work in progress, Craig’s paper is missing some of the niceties of a more polished academic work, like good referencing and a thoroughly researched introduction that places the work in context (the most notable omission is the lack of background material on dimensional analysis, which I will talk about in this post). Despite its rough edges, Craig’s paper led me down quite an interesting rabbit-hole, of which I hope to give you a glimpse. This post covers some background concepts; I’ll mention Craig’s contribution in a follow-up post. ]

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Imagine you have just woken up after a very bad hangover. You retain your basic faculties, such as the ability to reason and speak, but you have forgotten everything about the world in which you live. Not just your name and address, but your whole life history, family and friends, and entire education are lost to the epic blackout. Using pure thought, you are nevertheless able to deduce some facts about the world, such as the fact that you were probably drinking Tequila last night.

RM_hangover
The first thing you notice about the world around you is that it can be separated into objects distinct from yourself. These objects all possess properties: they have colour, weight, smell, texture. For instance, the leftover pizza is off-yellow, smells like sardines and sticks to your face (you run to the bathroom).

While bending over the toilet for an extended period of time, you notice that some properties can be easily measured, while others are more intangible. The toilet seems to be less white than the sink, and the sink less white than the curtains. But how much less? You cannot seem to put a number on it. On the other hand, you know from the ticking of the clock on the wall that you have spent 37 seconds thinking about it, which is exactly 14 seconds more than the time you spent thinking about calling a doctor.

You can measure exactly how much you weigh on the bathroom scale. You can also see how disheveled you look in the mirror. Unlike your weight, you have no idea how to quantify the amount of your disheveled-ness. You can say for sure that you are less disheveled than Johnny Depp after sleeping under a bridge, but beyond that, you can’t really put a number on it. Properties like time, weight and blood-alcohol content can be quantified, while other properties like squishiness, smelliness and dishevelled-ness are not easily converted into numbers.

You have rediscovered one of the first basic truths about the world: all that we know comes from our experience, and the objects of our experience can only be compared to other objects of experience. Some of those comparisons can be numerical, allowing us to say how much more or less of something one object has than another. These cases are the beginning of scientific inquiry: if you can put a number on it, then you can do science with it.

Rulers, stopwatches, compasses, bathroom scales — these are used as reference objects for measuring the `muchness’ of certain properties, namely, length, duration, angle, and weight. Looking in your wallet, you discover that you have exactly 5 dollars of cash, a receipt from a taxi for 30 dollars, and you are exactly 24 years old since yesterday night.

You reflect on the meaning of time. A year means the time it takes the Earth to go around the Sun, or approximately 365 and a quarter days. A day is the time it takes for the Earth to spin once on its axis. You remember your school teacher saying that all units of time are defined in terms of seconds, and one second is defined as 9192631770 oscillations of the light emitted by a Caesium atom. Why exactly 9192631770, you wonder? What if we just said 2 oscillations? A quick calculation shows that this would make you about 110 billion years old according to your new measure of time. Or what about switching to dog years, which are 7 per human year? That would make you 168 dog years old. You wouldn’t feel any different — you would just be having a lot more birthday parties. Given the events of last night, that seems like a bad idea.

You are twice as old as your cousin, and that is true in dog years, cat years, or clown years [2]. Similarly, you could measure your height in inches, centimeters, or stacked shot-glasses — but even though you might be 800 rice-crackers tall, you still won’t be able to reach the aspirin in the top shelf of the cupboard. Similarly, counting all your money in cents instead of dollars will make it a bigger number, but won’t actually make you richer. These are all examples of passive transformations of units, where you imagine measuring something using one set of units instead of another. Passive transformations change nothing in reality: they are all in your head. Changing the labels on objects clearly cannot change the physical relationships between them.

Things get interesting when we consider active transformations. If a passive transformation is like saying the length of your coffee table is 100 times larger when measured in cm than when measured in meters, then an active transformation would be if someone actually replaced your coffee table with a table 100 times bigger. Now, obviously you would notice the difference because the table wouldn’t fit in your apartment anymore. But imagine that someone, in addition to replacing the coffee table, also replaced your entire apartment and everything in it with scaled-up models 100 times the size. And imagine that you also grew to into a giant 100 times your original size while you were sleeping. Then when you woke up, as a giant inside a giant apartment with a giant coffee table, would you realise anything had changed? And if you made yourself a giant cup of coffee, would it make your giant hangover go away?

Kafka
Or if you woke up as a giant bug?

We now come to one of the deepest principles of physics, called Bridgman’s Principle of absolute significance of relative magnitude, named for our old friend Percy Bridgman. The Principle says that only relative quantities can enter into the laws of physics. This means that, whatever experiments I do and whatever measurements I perform, I can only obtain information about the relative sizes of quantities: the length of the coffee table relative to my ruler, or the mass of the table relative to the mass of my body, etc. According to this principle, actively changing the absolute values of some quantity by the same proportion for all objects should not affect the outcomes of any experiments we could perform.

To get a feeling for what the principle means, imagine you are a primitive scientist. You notice that fruit hanging from trees tends to bob up and down in the wind, but the heavier fruits seems to bounce more slowly than the lighter fruits (for those readers who are physics students, I’m talking about a mass on a spring here). You decide to discover the law that relates the frequency of bobbing motion to the mass of the fruit. You fill a sack with some pebbles (carefully chosen to all have the same weight) and hang it from a tree branch. You can measure the mass of the sack by counting the number of pebbles in it, but you still need a way to measure the frequency of the bobbing. Nearby you hear the sound of water dripping from a leaf into a pond. You decide to measure the frequency by how many times the sack bobs up and down in between drips of water. Now you are ready to do your experiment.

You measure the bobbing frequency of the sack for many different masses, and record the results by drawing in the dirt with a stick. After analysing your data, you discover that the frequency f (in oscillations per water drop) is related to the mass m (in pebbles) by a simple formula:

HookeSpring
where k stands for a particular number, say 16.8. But what does this number really mean?

Unbeknownst to you, a clever monkey was watching you from the bushes while you did the experiment. After you retire to your cave to sleep, the monkey comes out to play a trick on you. He carefully replaces each one of your pebbles with a heavier pebble of the same size and appearance, and makes sure that all of the heavier pebbles are the same weight as each other. He takes away the original pebbles and hides them. The next day, you repeat the experiment in exactly the same way, but now you discover that the constant k has changed from yesterday’s value of 16.8 to the new value of 11.2. Does this mean that the law of nature that governs the bobbing of things hanging from the tree has changed overnight? Or should you decide that the law is the same, but that the units that you used to measure frequency and mass have changed?

You decide to apply Bridgman’s Principle. The principle says that if (say) all the masses in the experiment were changed by the same proportion, then the laws of physics would not allow us to see any difference, provided we used the same measuring units. Since you do see a difference, Bridgman’s Principle says that it must be the units (and not the law itself) that has changed. `These must be different pebbles’ you say to yourself, and you mark them by scratching an X onto them. You go out looking for some other pebbles and eventually you find a new set of pebbles which give you the right value of 16.8 when you perform the experiment. `These must be the same kind of pebbles that I used in the original experiment’ you say to yourself, and you scratch an O on them so that you won’t lose them again. Ha! You have outsmarted the monkey.

Larson_rocks

Notice that as long as you use the right value for k — which depends on whether you measure the mass using X or O pebbles — then the abstract equation (1) remains true. In physics language, you are interpreting k as a dimensional constant, having the dimensions of  frequency times √mass. This means that if you use different units for measuring frequency or mass, the numerical value of k has to change in order to preserve the law. Notice also that the dimensions of k are chosen so that equation (1) has the same dimensions on each side of the equals sign. This is called a dimensionally homogeneous equation. Bridgman’s Principle can be rephrased as saying that all physical laws must be described by dimensionally homogeneous equations.

Bridgman’s Principle is useful because it allows us to start with a law expressed in particular units, in this case `oscillations per water-drop’ and `O-pebbles’, and then infer that the law holds for any units. Even though the numerical value of k changes when we change units, it remains the same in any fixed choice of units, so it represents a physical constant of nature.

The alternative is to insist that our units are the same as before (the pebbles look identical after all). That means that the change in k implies a change in the law itself, for instance, it implies that the same mass hanging from the tree today will bob up and down more slowly than it did yesterday. In our example, it turns out that Bridgman’s Principle leads us to the correct conclusion: that some tricky monkey must have switched our pebbles. But can the principle ever fail? What if physical laws really do change?

Suppose that after returning to your cave, the tricky monkey decides to have another go at fooling you. He climbs up the tree and whispers into its leaves: `Do you know why that primitive scientist is always hanging things from your branch? She is testing how strong you are! Make your branches as stiff and strong as you can tomorrow, and she will reward you with water from the pond’.

The next day, you perform the experiment a third time — being sure to use your `O-pebbles’ this time — and you discover again that the value of k seems to have changed. It now takes many more pebbles to achieve a given frequency than it did on the first day. Using Bridgman’s Principle, you again decide that something must be wrong with your measuring units. Maybe this time it is the dripping water that is wrong and needs to be adjusted, or maybe you have confidence in the regularity of the water drip and conclude that the `O-pebbles’ have somehow become too light. Perhaps, you conjecture, they were replaced by the tricky monkey again? So you throw them out and go searching for some heavier pebbles. You find some that give you the right value of k=16.8, and conclude that these are the real `O-pebbles’.

The difference is that this time, you were tricked! In fact the pebbles you threw out were the real `O-pebbles’. The change in k came from the background conditions of the experiment, namely the stiffness in the tree branches, which you did not consider as a physical variable. Hence, in a sense, the law that relates bobbing frequency to mass (for this tree) has indeed changed [3].

You thought that the change in the constant k was caused by using the wrong measuring units, but in fact it was due to a change in the physical constant k itself. This is an example of a scenario where a physical constant turns out not to be constant after all. If we simply assume Bridgman’s Principle to be true without carefully checking whether it is justified, then it is harder to discover situations in which the physical constants themselves are changing. So, Bridgman’s Principle can be thought of as the assumption that the values of physical constants (expressed in some fixed units) don’t change over time. If we are sure that the laws of physics are constant, then we can use the Principle to detect changes or inaccuracies in our measuring devices that define the physical units — i.e. we can leverage the laws of physics to improve the accuracy of our measuring devices.

We can’t always trust our measuring units, but the monkey also showed us that we can’t always trust the laws of physics. After all, scientific progress depends on occasionally throwing out old laws and replacing them with more accurate ones. In our example, a new law that includes the tree-branch stiffness as a variable would be the obvious next step.

One of the more artistic aspects of the scientific method is knowing when to trust your measuring devices, and when to trust the laws of physics [4]. Progress is made by `bootstrapping’ from one to the other: first we trust our units and use them to discover a physical law, and then we trust in the physical law and use it to define better units, and so on. It sounds like a circular process, but actually it represents the gradual refinement of knowledge, through increasingly smaller adjustments from different angles. Imagine trying to balance a scale by placing handfuls of sand on each side. At first you just dump about a handful on each side and see which is heavier. Then you add a smaller amount to the lighter side until it becomes heavier. Then you add an even smaller amount to the other side until it becomes heavier, and so on, until the scale is almost perfectly balanced. In a similar way, switching back and forth between physical laws and measurement units actually results in both the laws and measuring instruments becoming more accurate over time.

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[1] It is a shame that Craig’s work remains incomplete, because I think physicists could benefit from a re-examination of the principles of dimensional analysis. Simplified dimensional arguments are sometimes invoked in the literature on quantum gravity without due consideration for their meaning.

[2] Clowns have several birthdays a week, but they aren’t allowed to get drunk at them, which kind of defeats the purpose if you ask me.

[3] If you are uncomfortable with treating the branch stiffness as part of the physical law, imagine instead that the strength of gravity actually becomes weaker overnight.

[4] This is related to a deep result in the philosophy of science called the Duhem-Quine Thesis.
Quoth Duhem: `If the predicted phenomenon is not produced, not only is the questioned proposition put into doubt, but also the whole theoretical scaffolding used by the physicist’.

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Bootstrapping to quantum gravity

Kepler

“If … there were no solid bodies in nature there would be no geometry.”
-Poincaré

A while ago, I discussed the mystery of why matter should be the source of gravity. To date, this remains simply an empirical fact. The deep insight of general relativity – that gravity is the geometry of space and time – only provides us with a modern twist: why should matter dictate the geometry of space-time?

There is a possible answer, but it requires us to understand space-time in a different way: as an abstraction that is derived from the properties of matter itself. Under this interpretation, it is perfectly natural that matter should affect space-time geometry, because space-time is not simply a stage against which matter dances, but is fundamentally dependent on matter for its existence. I will elaborate on this idea and explain how it leads to a new avenue of approach to quantum gravity.

First consider what we mean when we talk about space and time. We can judge how far away a train is by listening to the tracks, or gauge how deep a well is by dropping a stone in and waiting to hear the echo. We can tell a mountain is far away just by looking at it, and that the cat is nearby by tripping over it. In all these examples, an interaction is necessary between myself and the object, sometimes through an intermediary (the light reflected off the mountain into my eyes) and sometimes not (tripping over the cat). Things can also be far away in time. I obviously cannot interact with people who lived in the past (unless I have a time machine), or people who have yet to be born, even if they stood (or will stand) exactly where I am standing now. I cannot easily talk to my father when he was my age, but I can almost do it, just by talking to him now and asking him to remember his past self. When we say that something is far away in either space or time, what we really mean is that it is hard to interact with, and this difficulty of interaction has certain universal qualities that we give the names `distance’ and `time’.
It is worth mentioning here, as an aside, that in a certain sense, the properties of `time’ can be reduced to properties of `distance’ alone. Consider, for instance, that most of our interactions can be reduced to measurements of distances of things from us, at a given time. To know the time, I invariably look at the distance the minute hand has traversed along its cycle on the face of my watch. Our clocks are just systems with `internal’ distances, and it is the varying correspondence of these `clock distances’ with the distances of other things that we call the `time’. Indeed, Julian Barbour has developed this idea into a whole research program in which dynamics is fundamentally spatial, called Shape Dynamics.

Sigmund Freud Museum, Wien – Peter Kogler

So, if distance and time is just a way of describing certain properties of matter, what is the thing we call space-time?

We now arrive at a crucial point that has been stressed by philosopher Harvey Brown: the rigid rods and clocks with which we claim to measure space-time do not really measure it, in the traditional sense of the word `measure’. A measurement implies an interaction, and to measure space-time would be to grant space-time the same status as a physical body that can be interacted with. (To be sure, this is exactly how many people do wish to interpret space-time; see for instance space-time substantivalism and ontological structural realism).

Brown writes:
“One of Bell’s professed aims in his 1976 paper on `How to teach relativity’ was to fend off `premature philosophizing about space and time’. He hoped to achieve this by demonstrating with an appropriate model that a moving rod contracts, and a moving clock dilates, because of how it is made up and not because of the nature of its spatio-temporal environment. Bell was surely right. Indeed, if it is the structure of the background spacetime that accounts for the phenomenon, by what mechanism is the rod or clock informed as to what this structure is? How does this material object get to know which type of space-time — Galilean or Minkowskian, say — it is immersed in?” [1]

I claim that rods and clocks do not measure space-time, they embody space-time. Space-time is an idealized description of how material rods and clocks interact with other matter. This distinction is important because it has implications for quantum gravity. If we adopt the more popular view that space-time is an independently existing ontological construct, it stands to reason that, like other classical fields, we should attempt to directly quantise the space-time field. This is the approach adopted in Loop Quantum Gravity and extolled by Rovelli:

“Physical reality is now described as a complex interacting ensemble of entities (fields), the location of which is only meaningful with respect to one another. The relation among dynamical entities of being contiguous … is the foundation of the space-time structure. Among these various entities, there is one, the gravitational field, which interacts with every other one and thus determines the relative motion of the individual components of every object we want to use as rod or clock. Because of that, it admits a metrical interpretation.” [2]

One of the advantages of this point of view is that it dissolves some seemingly paradoxical features of general relativity, such as the fact that geometry can exist without (non-gravitational) matter, or the fact that geometry can carry energy and momentum. Since gravity is a field in its own right, it doesn’t depend on the other fields for its existence, nor is there any problem with it being able to carry energy. On the other hand, this point of view tempts us into framing quantum gravity as the mathematical problem of quantising the gravitational field. This, I think, is misguided.

I propose instead to return to a more Machian viewpoint, according to which space-time is contingent on (and not independent of) the existence of matter. Now the description of quantum space-time should follow, in principle, from an appropriate description of quantum matter, i.e. of quantum rods and clocks. From this perspective, the challenge of quantum gravity is to rebuild space-time from the ground up — to carry out Einstein’s revolution a second time over, but using quantum material as the building blocks.

Ernst Mach vs. Max Ernst. Get it right, folks.

My view about space-time can be seen as a kind of `pulling oneself up by one’s bootstraps’, or a Wittgenstein’s ladder (in which one climbs to the top of a ladder and then throws the ladder away). It works like this:
Step 1: define the properties of space-time according to the behaviour of rods and clocks.
Step 2: look for universal patterns or symmetries among these rods and clocks.
Step 3: take the ideal form of this symmetry and promote it to an independently existing object called `space-time’.
Step 4: Having liberated space-time from the material objects from which it was conceived, use it as the independent standard against which to compare rods and clocks.

Seen in this light, the idea of judging a rod or a clock by its ability to measure space or time is a convenient illusion: in fact we are testing real rods and clocks against what is essentially an embodiment of their own Platonic ideals, which are in turn conceived as the forms which give the laws of physics their most elegant expression. A pertinent example, much used by Julian Barbour, is Ephemeris time and the notion of a `good clock’. First, by using material bodies like pendulums and planets to serve as clocks, we find that the motions of material bodies approximately conform to Newton’s laws of mechanics and gravitation. We then make a metaphysical leap and declare the laws to be exactly true, and the inaccuracies to be due to imperfections in the clocks used to collect the data. This leads to the definition of the `Ephemeris time’, the time relative to which the planetary motions conform most closely to Newton’s laws, and a `good clock’ is then defined to be a clock whose time is closest to Ephemeris time.

The same thing happens in making the leap to special relativity. Einstein observed that, in light of Maxwell’s theory of electromagnetism, the empirical law of the relativity of motion seemed to have only a limited validity in nature. That is, assuming no changes to the behaviour of rods and clocks used to make measurements, it would not be possible to establish the law of the relativity of motion for electrodynamic bodies. Einstein made a metaphysical leap: he decided to upgrade this law to the universal Principle of Relativity, and to interpret its apparent inapplicability to electromagnetism as the failure of the rods and clocks used to test its validity. By constructing new rods and clocks that incorporated electromagnetism in the form of hypothetical light beams bouncing between mirrors, Einstein rebuilt space-time so as to give the laws of physics a more elegant form, in which the Relativity Principle is valid in the same regime as Maxwell’s equations.

Ladder for Booker T. Washington – Martin Puryear

By now, you can guess how I will interpret the step to general relativity. Empirical observations seem to suggest a (local) equivalence between a uniformly accelerated lab and a stationary lab in a gravitational field. However, as long as we consider `ideal’ clocks to conform to flat Minkowski space-time, we have to regard the time-dilated clocks of a gravitationally affected observer as being faulty. The empirical fact that observers stationary in a gravitational field cannot distinguish themselves (locally) from uniformly accelerated observers then seems accidental; there appears no reason why an observer could not locally detect the presence of gravity by comparing his normal clock to an `ideal clock’ that is somehow protected from gravity. On the other hand, if we raise this empirical indistinguishability to a matter of principle – the Einstein Equivalence Principle – we must conclude that time dilation should be incorporated into the very definition of an `ideal’ clock, and similarly with the gravitational effects on rods. Once the ideal rods and clocks are updated to include gravitational effects as part of their constitution (and not an interfering external force) they give rise to a geometry that is curved. Most magically of all, if we choose the simplest way to couple this geometry to matter (the Einstein Field Equations), we find that there is no need for a gravitational force at all: bodies follow the paths dictated by gravity simply because these are now the inertial paths followed by freely moving bodies in the curved space-time. Thus, gravity can be entirely replaced by geometry of space-time.

As we can see from the above examples, each revolution in our idea of space-time was achieved by reconsidering the nature of rods and clocks, so as to make the laws of physics take a more elegant form by incorporating some new physical principle (eg. the Relativity and Equivalence principles). What is remarkable is that this method does not require us to go all the way back to the fundamental properties of matter, prior to space-time, and derive everything again from scratch (the constructive theory approach). Instead, we can start from a previously existing conception of space-time and then upgrade it by modifying its primary elements (rods and clocks) to incorporate some new principle as part of physical law (the principle theory approach). The question is, will quantum gravity let us get away with the same trick?

I’m betting that it will. The challenge is to identify the empirical principle (or principles) that embody quantum mechanics, and upgrade them to universal principles by incorporating them into the very conception of the rods and clocks out of which general relativistic space-time is made. The result will be, hopefully, a picture of quantum geometry that retains a clear operational interpretation. Perhaps even Percy Bridgman, who dismissed the Planck length as being of “no significance whatever” [3] due to its empirical inaccessibility, would approve.

Boots with laces – Van Gogh

[1] Brown, Physical Relativity, p8.
[2] Rovelli, `Halfway through the woods: contemporary research on space and time’, in The Cosmos of Science, p194.
[3] Bridgman, Dimensional Analysis, p101.

So can we time-travel or not?!

In a comment on my last post, elkement asked:

“What are exactly are the limits for having an object time-travel that is a bit larger than a single particle? Or what was the scope of your work? I am asking because papers as your thesis are very often hyped in popular media as `It has been proven that time-travel does work’ (Insert standard sci-fi picture of curved space here). As far as I can decode the underlying papers such models are mainly valid for single particles (?) but I have no feeling about numbers and dimensions, decoherence etc.”

Yep, that is pretty much THE question about time travel – can we do it with people or not? (Or even with rats, that would be good too). The bottom line is that we still don’t know, but I might as well give a longer answer, since it is just interesting enough to warrant its own blog post.

First of all, nobody has yet been able to prove that time travel is either possible or impossible according to the laws of physics. This is largely because we don’t yet know what laws govern time travel — for that we’d almost certainly need a theory of quantum gravity. In order for humans to time-travel, we would probably need to use a space-time wormhole, as proposed by Morris, Thorne and Yurtsever in the late eighties [1]. Their paper originated the classic wormhole graphic that everyone is so fond of:

(Used without permission from the publisher -- shh!)
(Used without permission from the publisher — shh!)

However, there are at least a couple of compelling arguments why it should be impossible to send people back in time, one of which is Stephen Hawking’s “Chronology Protection Conjecture”. This is commonly misrepresented as the argument “if time travel is possible, where are all the tourists from the future?”. While Stephen Hawking did make a comment along these lines, he was just joking around. Besides, there is a perfectly good reason why we might not have been visited by travellers from the future: according to the wormhole model, you can only go back in time as far as the moment when you first invented the time machine, or equivalently, the time at which the first wormhole mouth opens up. Since we haven’t found any wormhole entrances in space, nor have we created one artificially, it is no surprise that we haven’t received any visitors from the future.

via Saturday Morning Breakfast Cereal
via Saturday Morning Breakfast Cereal

The real Chronology Protection Conjecture involves a lot more mathematics and head-scratching. Basically, it says that matter and energy should accumulate near the wormhole entrance so quickly that the whole thing will collapse into a black hole before anybody has time to travel through it. The reason that it is still only a conjecture and has not been proven, is that it relies upon certain assumptions about quantum gravity that may or may not be true — we won’t know until we have such a theory. And then it might just turn out that the wormhole is somehow stable after all.

The other reason why time travel for large objects might be impossible is that, in order for the wormhole to be stable and not collapse in on itself Hawking-style, you need matter with certain quantum properties that can support the wormhole against collapse [2]. But it might turn out that it is just impossible to create enough of this special matter in the vicinity of a wormhole to keep it open. This is a question that one could hope to answer without needing a full theory of quantum gravity, because it depends only on the shape of the space-time and certain properties of quantum fields within that space-time. However, the task of answering this question is so ridiculously difficult mathematically that nobody has yet been able to do it. So the door is still open to the possibility of time-travelling humans, at least in theory.

To my mind, though, the biggest reason is not theoretical but practical: how the heck do you create a wormhole? We can’t even create a black hole of any decent size (if any had shown up at the LHC they would have been microscopic and very short-lived). So how can we hope to be able to manipulate the vast amounts of matter and energy required to bend space-time into a loop (and a stable loop no less), without annihilating ourselves in the process? Even if we were lucky to find a big enough, ready-made wormhole somewhere out in space, it will almost certainly be so far away as to make it nearly impossible to get there, due to sheer demands on technology. It’s a bit like asking, could humans ever build a friendly hotel in the centre of the sun? Well, it might be technically possible, but there is no way it would ever happen; even if we could raise humungous venture capital for the Centre-of-the-Sun Hotel, it would just be too damn hard.

The good news is that it might be more feasible to create a cute, miniature wormhole that only exists for a short time. This would require much smaller energies that might not destroy us in the process, and might be easier to manipulate and control (assuming quantum gravity allows it at all). So, while there is as yet no damning proof that time-travel is impossible, I still suspect that the best we can ever hope to do is to be able to send an electron back in time by a very short amount, probably not more than one millisecond — which would be exciting for science nerds, but perhaps not the headline that the newspapers would have wanted.

[1] Fun fact: while working on the movie  “Contact”, Carl Sagan consulted Kip Thorne about the physics of time-travel.

[2] For the nerds out there, you need matter that violates the averaged null energy condition (ANEC). You can look up what this means in any textbook on General Relativity — for example this one.

My PhD thesis on time-travel, for dummies.

One of the best things about the internet is that it gives the public unprecedented access to the cutting edge of science. For example, on Friday I uploaded my entire PhD thesis, “Causality Violation and Nonlinear Quantum Mechanics”, to the arXiv, where it can now be read by my contemporaries, colleagues, parents, neighbours and any pets that can read human-speak. The downside is, as my parents pointed out, my thesis is not written in English but in gobbledegook that makes no sense to anybody outside of the quantum physics community. So, while most work at the cutting edge of physics is available to anyone to read, almost nobody can understand it — and those who can, have access to journal subscriptions anyway! Oh, the irony. In order to remedy this problem, I’ve decided to translate my thesis from jargon into words that most people can understand (I’m not the first one to have this idea – check out Scott Aaronson’s description of his own research using the 1000 most common words in English).

Like any PhD thesis, my work is on a really small, pigeonhole topic that is part of a much bigger picture — like one brick in the wall of a huge building that is being constructed. I’ll start by talking about the whole building, which represents all of physics, and then slowly zoom in on the single, tiny brick that I focused on intensely for three and a half years. One thing about doing a PhD is that it teaches you to be very Zen about your work. We can’t all expect to be like Einstein, who single-handedly constructed an entire wing of the building — us regular scientists have to content ourselves with the laying of a single brick. In the end, the whole structure will only stand if every single brick is placed carefully and correctly. Sherlock Holmes once observed that, from a single drop of water, “a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. So all life is a great chain, the nature of which is known whenever we are shown a single link of it.” In the same way, the contribution of just a single brick to the building of physics is significant. After all, one cannot know how to place the individual brick without having first consulted the blueprint of the entire building. In this sense, the individual brick is as significant as the entire structure.

But enough philosophical mumbo-jumbo! Let’s get concrete. Modern physics can be roughly divided up into two separate theories. The first is Einstein’s theory of gravity, called General Relativity. This theory applies to all heavy objects that are affected by gravity, like planets, galaxies, and your mum. The theory says that heavy objects cause space and time to bend and curve around them in a special way. When the objects move through this curved space-time, they follow paths that bring them closer together, creating the appearance of an attractive force — the “force” normally known as gravity. If you want to fly rockets through space or deduce how the galaxy formed, or predict how heavy objects move around each other in space, you need this theory.

The other theory is Quantum Mechanics. This governs the behaviour of things that are very small, like atoms and the smaller parts of atoms. It turns out that even light is composed of a vast number of very small particles, called photons. If you have a very weak source of light that emits only a few photons at a time, it too is described by quantum mechanics. All quantum particles possess wave-like behaviour under certain conditions, which is described by something called the “wave-function” of the particle. Since we are made entirely out of atoms, you might ask: how come our bodies are not governed by quantum mechanics too? The answer is that, although in theory there could be a wave-function for larger objects like human bodies, in practice it is extremely difficult to create the special conditions needed to see the quantum behaviour of such large collections of atoms. The process whereby quantum effects become harder to observe in bigger objects is called “decoherence”, and it is still not completely understood.

Now comes the interesting part: it turns out that gravity is very weak. We can feel the Earth’s gravitational pull because the Earth is really, really heavy. But, in theory, you and I are also heavy enough to cause space-time to bend, just a little bit, around our bodies. This means you should be pulled towards me by gravity, and I towards you. However, because we are not nearly as heavy as the Earth, our gravitational pull on each other is far too weak for us to notice (though people have observed the gravitational attraction between large, heavy balls suspended by wires). By the time you get down to molecules, atoms and quantum mechanics, the gravitational force is so tiny that it can be completely ignored. On the other hand, because of decoherence, quantum mechanics can usually be ignored for anything much larger than molecules. This means that it is very rare to find a situation in which both quantum effects and gravitational effects both need to be taken into account; it is almost always a case of just one or the other.

The trouble is, if there were an in-between situation where both theories become significant, we do not know how quantum mechanics and gravity would overlap. One example is when a star becomes so heavy that it collapses under its own gravity into a black hole. The singularity at the centre of the hole is small enough that quantum mechanics should become important. As for being heavy, well, a black hole is so heavy that even light cannot go fast enough to escape from its gravitational attraction. Another in-between case would occur if we ever manage to make instruments that can measure space and time to extremely high accuracy, at a level called the “Planck scale”. If we are ever able to measure such tiny distances and times, we might expect to observe the curving of space-time due to very low-mass objects like atoms. So far, such precision is beyond our technology, and black holes have only been observed very indirectly. But when we finally do begin to probe these things, we will need a theory to describe them that incorporates both gravity and quantum mechanics together — a theory of “quantum gravity”. But despite roughly a hundred years of effort, we still do not have such a theory.

Why is it so hard to do quantum mechanics and gravity at the same time? This question alone is the subject of much debate. For now, you’ll just have to take my word that you can’t simply mash the two together (it has something to do with the fact that “space-time” is no longer clearly defined at the Planck scale). One approach is to consider a specific example of something that needs quantum gravity to describe it, like a black hole, and then try to develop a kind of “toy” theory of quantum gravity that describes only that particular situation. Once you have enough toy theories for different situations, you might be able to stick them together into a proper theory that includes all of them as special cases. This is easier said than done — it relies on us being able to come up with a wide variety of “thought experiments” that combine different aspects of quantum mechanics and gravity. But thought experiments like these are very rare: you’ve got black holes, Roger Penrose’s idea of a massive object in a quantum superposition, and a smattering of lesser known ideas. Are there any more?

This is where I come in. The idea for my PhD thesis comes from a paper that I stumbled across as an undergraduate at the University of Melbourne. The paper, by Tim Ralph, Gerard Milburn and Tony Downes of the University of Queensland, proposed that Earth’s own gravitational field might be strong enough to cause quantum gravity effects in experiments done on satellites. In particular, the difference between the strength of gravity at ground-level and at the height of the orbiting satellite might be just enough to make the quantum particles on the satellite behave in a very funny “non-linear” way, never before seen at ground level. Why might this happen? This is where the story gets bizarre: the authors got their idea after looking at a theory of time-travel, proposed in 1991 by David Deutsch. According to Deutsch’s theory, if space and time were bent enough by gravity to create a closed loop in time (aka a time machine), then any quantum particle that travelled backwards in time ought to have a very peculiar “non-linear” behaviour. Tim Ralph and co-authors said: what if there was only a little bit of space-time curvature? Wouldn’t you still expect just a little bit of non-linear behaviour? And we can look for that in the curvature produced by the Earth, without even needing to build a time-machine!

I was so fascinated by this idea that I immediately wrote to Tim Ralph. After some discussions, I visited Brisbane for the first time to meet him, and soon afterwards I began my PhD at the Uni of Queensland under Tim’s supervision. My first task was to understand Deutsch’s model of time-travel in more detail. (Unfortunately, the idea of time-travel is notorious for attracting crackpots, so if you want to do legitimate research on the topic you have to couch it in jargon to convince your colleagues that you have not gone crazy — hence the replacement of “time-travel” with the more politically-correct term “causality violation” in the title of my thesis). David Deutsch is best known for being one of the founding fathers of the idea of a quantum computer. That gave him enough credibility amongst physicists that we were willing to listen to his more eccentric ideas on things like Everett’s many-worlds interpretation of quantum mechanics, and the quantum mechanics of time-travel. On the latter topic, Deutsch made the seminal observation that certain well-known paradoxes of time travel  — for instance, what happens if you go back in time and kill your past self before he/she enters the time machine — could be fixed by taking into account the wave-function of a quantum particle as it travels back in time.

xkcd
Source: xkcd.com

Roughly speaking, the closed loop in time causes the wave-function of the particle to loop back on itself, allowing the particle to “interact with itself” in a non-linear way. Just like Schrödinger’s cat can be both dead and alive at the same time in a quantum superposition, it is possible for the quantum particle to “kill it’s past self” and “not kill it’s past self” at the same time, thereby apparently resolving the paradox (although, you might be forgiven for thinking that we just made it worse…).

Here was a prime example of a genuine quantum gravity effect: space-time had to be curved in order to create the time-machine, but quantum mechanics had to be included to resolve the paradoxes! Could we therefore use this model as a new thought experiment for understanding quantum gravity effects? Luckily for me, there were a few problems with Deutsch’s model that still needed to be ironed out before we could really take seriously the experiment proposed by Ralph, Milburn and Downes. First of all, as I mentioned, Deutsch’s model introduced non-linear behaviour of the quantum particle. But many years earlier, Nicolas Gisin and others had argued that any non-linear effects of this type should allow you to send signals faster than light, which seemed to be against the spirit of Einstein’s theory of relativity. Secondly, Deutsch’s model did not take into account all of the quantum properties of the particle, such as the spread of the particle’s wave function in space and time. Without that, it remained unclear whether the non-linear behaviour should persist in Earth’s gravitational field, as Ralph and co-authors had speculated, or whether the space-time spread of the wave-function would some how “smear out” the non-linearity until it disappeared altogether.

In my thesis, I showed that it might be possible to keep the non-linear behaviour of the Deutsch model while also ensuring that no signals could be sent faster than light outside of the time-machine. My arguments were based on some work that had already been done by others in a different context — I was able to adapt their work to the particular case of Deutsch’s model. In addition, I re-formulated Deutsch’s model to describe what happens to pulses of light, such that the space-time spread of the light could be taken into account together with the other quantum properties of light. Using this model, I showed that even if a pulse of light were sent back in time without interacting with its past self at all (no paradoxes), the wave-function of the light would still behave in a non-linear way. Using my model, I was able to describe exactly when the non-linear effects would get “smeared out” by the wave-function, and confirmed that the non-linear effects might still be observable in Earth’s gravitational field without needing a time-machine, thereby lending further support to the speculative work of Ralph and co. that had started it all.

So, that’s my PhD in a nutshell! Where to next? Right now I have decided to calm down a little bit and steer towards less extreme examples of a quantum gravity thought experiments. In particular, rather than looking at outright “causality violation”, I am investigating a peculiar effect called “indefinite causality”, in which space-time is not quite curved enough to send anything backwards in time, but where it is also not clear which events are causes and which ones are their effects. Hopefully, I’ll be able to understand how quantum mechanics fits into this weird picture — but that’s a topic for another post.

Is physics in a crisis?

We live in very interesting times, especially if you are a theoretical physicist like me. To understand what kind of time we are living in physics-wise, it will be helpful to review some ideas of Thomas Kuhn, a famous philosopher of science. Kuhn described science as proceeding through a series of paradigms. A `paradigm’ is a sort of established framework in which scientists work to solve problems using an agreed-upon set of tools. The paradigm provides both the puzzles to be solved and the tools to solve them. Over time, scientists discover that the tools of the paradigm cannot solve every puzzle. The problems that lie beyond the reach of a paradigm are called anomalies. When enough serious anomalies are discovered, scientists begin to lose confidence in the existing paradigm and a crisis occurs. Historically, each crisis has been resolved by a subsequent scientific revolution, in which the old paradigm was replaced by a new paradigm that is capable of resolving the anomalies [1].

Interestingly, although the new paradigm solves more problems than the old paradigm, it also represents a complete change in perspective, so that even those problems that were solved by the old paradigm have to be `re-solved’ by the new paradigm, from a completely new point of view. As a result, there might be the odd puzzle that was solved by the old paradigm but suddenly cannot be solved by the new paradigm! This phenomenon is known as `Kuhn-loss’. The new paradigm is successful so long as it solves more important puzzles than the ones it loses through Kuhn-loss. I mention this only to illustrate how significant a change in paradigm is from Kuhn’s point of view: it is not merely a period of accelerated science, but a complete reworking of how scientists see the world.

We are currently in a period of crisis. Some physicists might disagree with me, but I think one can make a strong case that the paradigm that has taken us this far is showing cracks. In this post, I won’t directly compare current events to Kuhn’s description of a crisis, nor will I spend effort trying to define what the present paradigm is. For the moment I will content myself by pointing to some (just a few!) of the major puzzles that are facing us, and explain why they may represent `anomalies’ that require a new paradigm in order to solve them [2].

Dark matter / energy: One of the best-known puzzles of our time is the mystery of dark matter and dark energy in cosmology. Briefly, the matter that we can see in the universe (galaxies, nebulae and so on) is moving around as though it is being pushed and pulled by gravitational forces that have no visible source. In fact, there seems to be 95% more `stuff’ in the universe that we can’t actually see directly – we can only deduce its presence by its gravitational interactions with visible matter. The fact that we don’t know what this stuff is has been called the most embarrassing problem in physics for good reason: if somebody asks me what kind of matter and energy there is in the universe, I have to admit that, for the most part, I have no freaking idea.

Quantum gravity: Going by Kuhn’s picture of science, the key tool of the present paradigm is the Standard Model (SM) of particle physics. This model is impressively accurate down to really tiny scales and has been spectacularly confirmed time and time again in the world’s big particle accelerators, right up to the recent discovery of the Higgs Boson at the Large Hadron Collider (LHC). However, a major limitation of the Standard Model is that it does not tell us how gravity fits into the picture. While we have brought electromagnetism and the nuclear forces up to date with quantum mechanics, our theory of gravity is still straggling behind by over a hundred years. All the other forces have been given a quantum makeover, but gravity remains the shy stepsister, cloaked in a classical veil. Despite some pioneering attempts to get behind that veil, most notably String Theory and Loop Quantum Gravity, there is still no agreement among the community about which approach is correct or whether we have to try something else entirely [3].

Quantum foundations: It is often said that nobody understands quantum mechanics. This would be very worrying if it were true, since much of today’s technology is based on it! So what is the situation really? Well, obviously we understand the theory well enough to use it in practical applications. The trouble is more on the philosophical side: physicists can’t agree on why quantum mechanics works so well. In fact, we still can’t agree on why the universe should be quantum mechanical in the first place! John Wheeler’s famous question `why the quantum?’ still keeps many of us awake at night. There is an ongoing body of research on quantum foundations, whose goal is to improve our understanding of quantum mechanics to the point where most of us can agree on a single interpretation. This interpretation (it is hoped) would reveal quantum mechanics in such a way that nearly every physicist will reflexively slap their forehead and declare `of course! It had to be that way’! The interpretation should be so compelling that classical physics will look absurd by comparison and quantum mechanics will be the most natural way to describe the world.

As an example, since Einstein, the gravitational force is now widely interpreted as the curvature of space and time. However, technically it is possible to explain gravity in terms of fields operating in flat spacetime, in a way that agrees with current experimental data – yet if you ask any physicist what gravity is, nearly all of them will say `the curvature of space-time due to matter’. By contrast, if you ask them what the wave function of quantum mechanics is, you will get all kinds of different answers, and probably an invitation to a conference on foundations where such matters are still being hotly debated. Whereas curved space-time seems like an elegant, simple and compelling way of visualizing gravity, we have no similarly compelling paradigm for visualizing quantum mechanics.

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One of the tasks a physicist faces during a crisis is to identify which anomalies deserve our attention and which ones are less important. This decision is guided by one’s intuitions and one’s chosen philosophy, hence a physicist must embrace some philosophy in order to make progress. For my part, I am most interested in the latter two anomalies: quantum gravity and quantum foundations. I think that the two are deeply connected. Since the regime of quantum gravity is still far from being accessible to experiments, the success of a theory of quantum gravity will be decided by the intuitive appeal of the physical principles on which it is based, as well as its elegance and explanatory power. We cannot hope to meet these demands all the way down at the level of quantum gravity (the Planck scale) if we still can’t do it up here on our home turf for quantum mechanics. Indeed, it is embarrassing that we cannot claim to have such a compelling picture of quantum mechanics, given that we have so much experimental data to guide us!

In upcoming blog posts I intend to elaborate on quantum gravity and quantum foundations and their possible connection to one another. I will also present my own ideas about how we should try to resolve the connected anomalies, using a philosophy based on a modern revival of operationalism and ideas from the exciting new field of quantum information. Stay tuned!

[1] This is a very rough version of Kuhn’s picture of scientific progress. The reader is encouraged to read the entry on Thomas Kuhn in the Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/thomas-kuhn/ . The less lazy reader is referred to Kuhn’s seminal work The Structure of Scientific Revolutions, University of Chicago Press, 2nd ed. (1970).

[2] There are of course far more anomalies in physics than the three listed here, although many of them can be linked to the same broad categories. For a more thorough list, see John Baez’s `Open Questions in Physics’: http://math.ucr.edu/home/baez/physics/General/open_questions.html .

[3] Some people have gone as far as to argue that String Theory is a failure. As an ignoramus, my own stance on this is more cautious, but that is a topic for another blog post.