Partially moving to Twitter

Posted in Uncategorized on April 3, 2016 by quantummoxie

It’s been six months since I posted anything. This is mostly because I lack the time to put serious work into blogging. However, I have finally decided to join the world of Twitter and so Quantum Moxie will mostly be moving over there, though I plan to post a few longer pieces here once in awhile. I’ve had bouts of success here but I think I can do more with Twitter than I can with this. Nevertheless, this site is not going away. Ideally I would love to figure out how to post to here from Twitter (rather than just have my Twitter feed embedded on the side). I’ll continue to work on that and I thank you (all two or three of you!) for your continued support!

The Weirdness of Neutrinos: beyond the 2015 Nobel Prize

Posted in Uncategorized on October 7, 2015 by quantummoxie

As most of you are probably aware, the Nobel Prize in Physics was awarded this week to Takaaki Kajita and Arthur B. McDonald “for the discovery of neutrino oscillations, which shows that neutrinos have mass”. This is truly a fantastic discovery and one that is long overdue for recognition. To be clear, what Kajita and McDonald won the award for was the experimental verification of neutrino oscillations. The theoretical prediction of such oscillations was made as early as 1957 by Bruno Pontecorvo. The Standard Model long predicted three types of neutrinos, but it also predicted that they should be massless (this is one of the glaring holes in the Standard Model that people too often ignore). In 1968 Pontecorvo figured out that if neutrinos actually had mass then they could change into one another. This was suggested as a solution to the so-called solar neutrino problem. The masses of the individual neutrinos were subsequently experimentally verified (and have led to a previous Nobel Prize in physics). What hadn’t been observed until Kajita and McDonald came along was the oscillation between the different types (masses), even though it was widely accepted that such oscillations existed.

The thing is, neutrinos are puzzling for another reason. Neutrinos are neutral particles and so don’t have an electric charge. Unlike neutrons, however, they are fundamental which means, as far as we know, they can’t be broken down into constituent particles. This fact is important for what I am about to say. All particles that possess angular momentum, also possess a magnetic dipole moment. So, for example, a dipole moment can arise from an electron’s orbit about a nucleus as well as its intrinsic spin. This is well-known, standard physics. Another piece of well-known, standard physics (Maxwell’s equations) tells us that magnetic fields are nothing more than electric fields viewed from a reference frame that is moving relative to the source of the electric field. This is why we typically refer to the singular field as “electromagnetic.” It’s just one field but it has two different (or seemingly different) behaviors depending on how it is viewed. The sources of such fields are electric charges. This actually helps explain the concept of magnetic moment. The presence of angular momentum in a system indicates the presence of relative motion between the portion of the system with the angular momentum and some observer. If the system possesses electric charge, then it makes sense that a magnetic field would be present as well due to the relative motion.

So now consider a neutron (not a neutrino just yet). It is a fermion and thus has a spin of 1/2 which suggests that it has a magnetic moment. In fact it does. But how does a neutral particle develop a magnetic moment if the source of magnetism ultimately has to be charge? Well, the standard way to answer this is to simply say that the neutron is a composite particle that is actually composed of quarks which are not neutral. So we could easily dismiss the magnetic moment of the neutron as being some kind of relic of the relative motion between an observer and the constituent quarks which do possess charge. But what about the neutrino?

This model would suggest that neutrinos shouldn’t have any magnetic moment at all because they are fundamental (i.e. they do not have any constituents). And yet, they do. But why? And how?

As it turns out, there is a connection between mass and magnetic moment according to ideas dating back to Dirac! Semi-classically, one can also show that any particle that possesses spin also possesses a magnetic dipole moment. Either way, however, since the Standard Model predicts a zero mass for the neutrino, any magnetic moment would suggest physics beyond the Standard Model. There are plenty of suggested minimal extensions of the Standard Model that produce a solution to this conundrum (e.g. you can use Feynman diagrams to show that there is a one-loop approximation of the neutrino as a “mixture” of a W+ and an e-). But none of these is universally accepted. And so we are left with another bizarre property of the neutrino: it is a neutral, fundamental particle that has electromagnetic characteristics! In short, it is the ultimate Standard Model contrarian.

A Short History of Quantum Reference Frames, Part 2

Posted in Uncategorized on July 18, 2015 by quantummoxie

Yesterday, I began outlining the history of quantum reference frames beginning with Marco Toller’s paper in 1977. What we saw was two, somewhat separate lines of development. The first was a direct line from Toller through Rovelli and back again to Toller, in which a generalized (and distinctly operational) notion of a reference frame is developed that covers both quantum and classical systems. The second was a line beginning with Holevo and leading up through Peres and Scudo, in which a protocol is developed for sending information about a Cartesian coordinate system via a quantum channel.

The question I have is, where do the lines converge? Specifically, when on the informational side do people start referring to reference frames in the more general sense? Pinpointing an exact instance is difficult, but an early paper by Bartlett, Rudolph, and Spekkens (hereafter referred to as BRS) in 2003 seems to suggest that Toller’s idea had made its way into the quantum information community by that time. The first few sentences of their paper read:

Quantum physics allows for powerful new communication tasks that are not possible classically, such as secure communication and entanglement-enhanced classical communication. In investigations of these and other communication tasks, considerable effort has been devoted to identifying the physical resources that are required for their implementation. It is generally presumed, at least implicitly, that a shared reference frame (SRF) between the communicating parties is such a resource, with the precise nature of the reference frame being dictated by the particular physical systems involved.

The emphasis on the word ‘physical’ in the second sentence is theirs. The emphasis in the last sentence is mine and meant to show that Toller’s idea had gained some traction in the QI community by then. In a meaty review article published in 2007, BRS offered a slightly different take on Toller’s notion of a reference frame (and notably still did not cite his work which makes me wonder if they were and still are unaware of it, or if they disagreed with his definition to some extent). They refer back to the work by Peres and Scudo and define two types of information: fungible and nonfungible (referred to by Peres and Scudo as ‘speakable’ and ‘unspeakable’ respectively). Fungible information is typically classical and refers to information for which the means of encoding does not matter. The example they give is that Shannon’s coding theorems don’t care one way or the other how the 0’s and 1’s are encoded (e.g. by magnetic tape or by voltages or some other manner). So this is fungible information. If, on the other hand, the encoding of information does make a difference to the information being encoded, then that information is said to be nonfungible (unspeakable). BRS then note that

[w]e refer to the systems with respect to which unspeakable/nonfungible information is defined, clocks, gyroscopes, metre sticks and so forth, as reference frames.

The emphasis is theirs. How does this compare to Toller’s definition of a frame of reference as some material object that is of the same nature as the objects that form the system under investigation, as well as the instruments used to measure that system? It seems to me that the BRS definition is merely a more concise statement of Toller’s definition which does not account for the differences between fungible and nonfungible information. So I think we’re still talking about the same thing here.

Superficially, it is fairly easy to see that many quantum ideas are natural generalizations of classical ideas. For example, in his terrific (and freely available!) undergraduate course materials on the mathematics of theoretical physics, Karl Svozil defines a reference frame (i.e. a “coordinate system”) as a linear basis. This is, arguably, a non-operational alternative to Toller’s definition, but I think the two can essentially be made equivalent. In fact he launches into a detailed discussion of the motivation behind defining the Cartesian reference frame which he refers to as “the Cartesian basis.” So the Cartesian frame of reference is just a special case of the broader idea of a degree of freedom as I pointed out in yesterday’s post. The point of all this is, even engineers will recognize that the even the classical notion of a reference frame is more than purely Cartesian (aerospace engineers, for example, work in systems with six degrees of freedom – either the three spatial coordinates plus the Euler angles [referred to by flight dynamics people as pitch, roll, and yaw], or the three spatial coordinates plus the corresponding momenta in each of those directions, i.e. phase space).

At any rate, one of the key results given in the BRS review has to do with the relation between quantum systems and classical systems. It is built on the notion of a superselection rule (SSR) which I briefly mentioned in yesterday’s post. SSRs were introduced by Wick, Wightman, and Wigner in 1952. In essence, they are rules that place limits on certain types of measurements, but are perhaps better understood as rules that prohibit superpositions of certain coherent states. What BRS showed in their review was that SSRs are formally equivalent to the lack of a shared classical reference frame (in the general sense of the term). So for instance, Wick, Wightman, and Wigner suggested that an SSR existed for states of opposite charge, i.e. one would never see a superposition of positive and negative charge. It makes perfect sense that this is equivalent to the lack of a classical reference frame for charge. In other words, we will never find a metal plate that is simultaneously positively charged and negatively charged. It is always one or the other or neutral. This might pose a problem for two parties (Alice and Bob again) who might, for some bizarre reason, have independently designed communication devices that rely on different charges. In a highly contrived example, suppose Bob lives on a planet that developed computers that run on positive charge somehow (maybe ionized atoms, for example), whereas Alice lives here on earth where our computers run on negative charge. If both use voltage readings to determine 0’s and 1’s in their binary codes, then they will get completely opposite results if they try to interpret each others’ machines. This is a completely classical issue.

On the other hand, at the quantum level, states exist that are a superposition of opposite charges despite the SSR (the first suggestion of this for charge was by Aharonov and Susskind in 1967 which is the paper that essentially launched our work on CPT symmetry). So while Alice and Bob may lack a classical reference frame for their computers, a common quantum reference frame can be established by using these superposition states as building blocks. Understanding this point really requires understanding precisely what a reference frame is as well as how it is used in communicating information between two parties. In other words, it requires understanding the difference between fungible and nonfungible information. It’s just another example of how an information theoretical view can shed light on some of the deepest (and seemingly uncontroversial) problems in physics.

A Short History of Quantum Reference Frames, Part 1

Posted in Uncategorized on July 17, 2015 by quantummoxie

I was recently at the Relativistic Quantum Information – North (RQI-N) meeting at Dartmouth College where I presented some of the work I have been doing on quantum reference frames and how to use them to overcome certain superselection rules (SSRs), specifically the SSR associated with CPT symmetry. I received a lot of terrific comments and numerous discussions have been spawned from the presentation. In particular, I seem to have introduced a number of people to the concept of a quantum reference frame for the first time. In fact I’m meeting back up with a few folks next week to discuss some of these issues. In preparing for next week’s discussions, I decided to do a little historical research to find out where the concept originated and how it got to its current form. I thought a blog post on the topic might be somewhat useful to some people out there and it would also provide me with a permanent record of some of the background papers in the field.

So what is a quantum reference frame? The idea appears to have first been proposed by Marco Toller in 1977 in a paper entitled “An operational analysis of the space-time structure” that appeared in Il Nuovo Cimento B. Here I quote the abstract:

We discuss the concepts related to space-time in a quantum-relativistic theory by means of the analysis of the physical procedures used to construct a new frame of reference starting from a pre-existent frame (transformation procedures). The physical objects which form a frame of reference are allowed to interact with the other physical objects and follow the laws of quantum physics. We suggest that there are conceptual limitations which do not permit the exact realization of a transformation of the Poincaré group by means of physical procedures. We remark also that the operations performed in order to construct a frame of reference perturb the surrounding physical objects and are influenced by them. We propose some general theoretical schemes which take these facts into account and permit the separation of the geometrical effects of a transformation procedure from the physical ones. Finally we find the conditions which permit the construction of a Poincaré-invariant theory of the usual kind by means of the introduction of some ideal concepts which have no direct operational meaning.

In other words, the most general definition of a ‘frame of reference’ is as some material object that is of the same nature as the objects that form the system under investigation as well as the measuring instruments themselves (Bohr’s classical-quantum contrast not withstanding). This idea was further developed by Aharonov and Kaufherr in 1984 in which they extended the principle of equivalence to quantum reference frames, and in a pair of articles written in 1991 by Carlo Rovelli (see here and here) which appear to have played some role in inspiring his relational interpretation of quantum mechanics. In this way, these ideas bear a striking resemblance to work attempted by Eddington in the 1930s and early 1940s (a topic I will leave for another blog post, but that served as the core topic of my long-forgotten PhD thesis).

Anyway, these ideas are clearly operational (Toller even uses the term in his original paper). They were, however, not necessarily informational, at least initially. However, in his 1982 book Probabilistic and Statistical Aspects of Quantum Theory, Alexander Holevo (who was just announced as the 2016 winner of the Claude E. Shannon award by the IEEE Information Theory Society) addressed the following question: can a system of N elementary spins (i.e. qubits, which weren’t yet named in 1982) be used to communicate, in a single transmission, the orientation of three mutually orthogonal unit vectors, i.e. a Cartesian reference frame? Holevo concluded that if the system had a well-defined total spin angular momentum J then, at best, only one of the three vectors could be communicated. A way around this limitation was found nearly two decades later by Bagan, Baig, and Muñoz-Tapia and, around the same time, Peres and Scudo found a way to do it with a single hydrogen atom. The idea was to allow two distant parties (i.e. our old friends Alice and Bob) to establish a common Cartesian reference frame simply using a quantum channel. Thus these papers, while informational in their focus, used the less general definition of a reference frame as a Cartesian coordinate system. In fact it is not entirely clear that any of these authors (or others working on similar ideas – see the previously mentioned paper by Bagan, et. al. for additional references) was aware of the more general definition of the reference frame originally proposed by Toller.

One of the key ideas in the early information-related papers was that the Cartesian frame, i.e. the concept of a spatial direction, could be encoded in a particle’s spin state. Somewhere along the line (it’s not quite clear to me yet exactly when) someone put these two ideas together and the more general concept of a quantum reference frame was born. It appears that somewhere around 2002 or 2003 someone realized that a spatial direction is an example of a degree of freedom. Of course even classical physicists – even many engineers – know that there are more general and abstract spaces that have more than three degrees of freedom (e.g. phase space). For decomposable systems, a distinction can be made between what might be called ‘collective’ degrees of freedom, i.e. those between a system and something external to it, and ‘relative’ degrees of freedom, i.e. those between the systems constituent parts. Several authors (including John Preskill, who was at RQI-N) recognized that encoding information into the collective degrees of freedom posed a number of problems. Beginning, to the best of my knowledge, in 1997 with a paper by Zanardi and Rasetti, encoding information into the relative degrees of freedom of a system was shown to be more optimal in some situations. Hopefully, you can see where this is headed. The relational degrees of freedom harken back to the general frame of reference á là Rovelli and his relational interpretation of QM. For example, take a look at this early paper by Bartlett, Rudolph, and Spekkens. The first few paragraphs offer a fairly nice summary of some of the work that had just recently come out on relative quantum information, though the paper itself still primarily deals with something spatial.

As early as 1996, Toller himself recognized that limitations in representations of the Poincaré group necessitated taking “internal” degrees of freedom into account when working with quantum reference frames. An example of such an internal degree of freedom is electrical charge. In fact, in our first paper in PRL, we introduce a new quantum number that represents all of the universally conserved internal quantum degrees of freedom (which happen to only be electrical charge and the difference between baryon number and lepton number), though we were unaware of Toller’s paper at that point (in fact I was unaware of it until I started working on this blog post). It may well be, in fact, that we are the first to have considered internal degrees of freedom in such a manner.

At any rate, in Part 2 of this short history, I will attempt to nail down exactly who first suggested using a generalized reference frame in the manner of Toller in an information communication scheme. I will then discuss the relation to SSRs which play a vitally important role in this story.

Objective reality (and the return of this blog)

Posted in Uncategorized on May 19, 2015 by quantummoxie

It has been a really, really, really long time since I posted anything. I had plenty to say but just an insane amount of stuff going on. Well, maybe that’s not quite right. It’s partially right, anyway. But I also couldn’t bear the thought of sitting down and writing. I’m a bit of an obsessive compulsive when it comes to writing. I’m one of those people who obsesses over every word and I was obsessing over a lot of other stuff so the thought of doing so for the blog just didn’t appeal to me for awhile. But now that the semester has ended and I have some free time, here I am.

So what is it that has me so inspired as to start clicking away again? Objective reality. Yes, indeed, folks I get that worked up over objective reality. It’s been a theme running through my life of late. Or, rather, it’s the denial of objective reality by other people that has gotten me riled up. George Orwell wrote a lot about objective reality. 1984 was particularly notable in that regard. One could argue that the Enlightenment was about objective reality. In fact, that’s exactly what it was about.

But I don’t have the time or energy to unload about the assault on objective reality that is modern society. (Plus, this is ostensibly a science blog.) Instead, what I do plan to do is to discuss the nature of objective reality in physics. The two greatest theories of twentieth century physics – relativity and quantum mechanics – both include subjective elements. Let’s talk about relativity first.

Technically speaking, relativity didn’t begin with Einstein. It actually goes back to Galileo and Newton. But time was still absolute. Einstein showed that both space and time were relative, i.e. subjective. Notably, special (and general) relativity supposedly rid us of an absolute reference frame. This idea was then coopted by non-physicists (and even by some physicists) to mean that there is no objective reality. But is that necessarily true? In fact, relativity provides us with a mechanism by which two observers can agree. That’s the entire point of the Lorentz transformations. In fact, that’s the idea behind the famous Twin Paradox. People often mistakenly think that the paradox is that one twin is older than the other. That’s actually not the real paradox. In fact, there really isn’t a paradox at all. One twin really is younger than the other. But maybe I’m getting ahead of myself.

The typical description of the Twin Paradox begins with (duh) twins. Let’s call our twins Alice and Bob. One twin (let’s say Bob) hops on a spacecraft, accelerates to a relativistic speed and travels off toward a distant star. Later, Bob comes back. Now, according to a superficial understanding of relativity, Alice will think that Bob is younger because he was moving at a relativistic speed, i.e. close to the speed of light. And time slows down the closer one gets to the speed of light so Alice will think Bob is younger. But – and here’s the apparent paradox – in Bob’s reference frame, it’s actually Alice – and the entire observable universe, in fact – that moves off at a relativistic speed while Bob remains stationary since we are always stationary in our own reference frames. So shouldn’t Alice appear younger to Bob? After all, everything’s relative, right?

Not quite. There are two things to consider here. The first is that, proper time is measured along worldlines (world lines) in relativity. The person with the longer worldline measures the shorter amount of proper time. So if we consider this problem from the reference frame of, say, the sun, the worldlines (and associated events for this scenario) would appear as in this diagram where time is measured in years (y) and space (distance) is measured in light-years (y).


Notice that Alice’s worldline just moves around and around the sun. Clearly Bob’s worldline is longer and hence he will be younger. This result is entirely objective because we stepped outside of the Alice-Bob system and observed it from a third reference frame. But what if we chose a fourth or a fifth reference frame? Could we not find one in which things looked the opposite? That brings us to the second thing we need to consider here.

There actually is something that is exactly the same in every inertial (i.e. non-accelerating) reference frame: the speed of light! This is not only an empirically provable result, it is one of the key axioms of special relativity. (As a cautionary note, special relativity only assumes that the speed of light should be a maximum speed; it does not predict an actual value for that speed.) So the speed of light is also an objective reality – it’s the same for everyone, everywhere! So the closer an object gets to the speed of light, the smaller the differences become between inertial reference frame observations of that object’s speed (with the exception of the object’s own frame in which it is always at rest). Thus, even if we switch to a fourth or a fifth inertial reference frame in order to observe Alice and Bob, because Bob is moving at highly relativistic speeds, we won’t see his worldline change much when we observe it in some other reference frame. Alice’s, on the other hand, might change dramatically. For example, we could be in a frame in which the solar system itself is moving at a relativistic speed in the +x direction. That would tilt Alice’s worldline quite a bit, but wouldn’t change Bob’s all that much (I encourage the interested reader to work out the details).

Now let’s turn to quantum mechanics. There are two somewhat related reasons that objective reality is questioned in quantum mechanics. First, as is well-known, standard (i.e. non-weak) measurements disturb the system being measured. As such, one could legitimately ask if we can ever truly know the state of a system since merely looking at it disturbs it. Second, measurement results in quantum mechanics depend on the context within which they are measured, a phenomenon known as contextuality. So, as an example, consider a sequence of three Stern-Gerlach devices. For those who are unfamiliar with such devices, they measure the spin angular momentum of a particle (in this case, a spin-1/2 particle) along a particular axis. The result of the measurement shows the particle to either be aligned or anti-aligned with the given axis. As shown in the following figure, whether it comes out of the top or bottom of a device, determines whether it is aligned or anti-aligned respectively.


The classical (neo-realist) assumption is that when we measure the spin along a particular axis, that result should be the same in any subsequent measurement on that axis regardless of any intermediate measurements (and assuming there are no other external interactions that affect the system in the interim). So, in other words, if A and C represent the same axis, one would expect that, for the figure shown above, the particle should exit from the top (+) of the last device. But this doesn’t happen in quantum systems. The outcome probabilities for a given spin measurement depend solely on the angle between the axis of the current measuring device and the axis of the device immediately preceding the current one. In the example shown, that means that the probabilities associated with where the particle will exit the last device, only depend on the angle between the second and third devices. Thus it doesn’t matter if A and C are the same axis since the first device is irrelevant to the outcome. In other words, it would seem that the result is entirely subjective since it is context-dependent.

But hang on a second. Sure, the spin measurement may be dependent on the context, but the actual nature of the particle never changes. For example, if you send a stream of electrons into such a sequence of devices you never see stream of protons coming out. It just doesn’t happen. So even though, for example, we may not know a state fully, since measurement results depend on context and since the act of measurement disturbs the system, there are certain things about systems that we do know with absolute certainty. These are things that everyone will agree on. They represent an objective reality.

The fact is, that’s precisely the point of science: it is a method whereby people can come to an agreement about an objective reality. We fully expect that an electron, when observed in New York, will be an electron when observed in Tokyo. We rely on that fact. And that’s one of the great lessons of relativity: the laws of physics must be the same in every inertial reference frame. We literally (not figuratively!) cannot imagine a universe in which they weren’t.

That brings me to my final point of this post. Supposedly, some QBists deny the existence of an objective reality. I’ve never actually gotten into a deep, philosophical discussion about reality with any of them. But I do know Carl Caves, Rüdiger Schack, and Chris Fuchs (the progenitors of QBism). Given some of the things Carl has said over the years in random conversations here and there, I would find it difficult to believe that he really didn’t believe in some kind of objective reality (and I plan to ask him the next time I talk to him). Rüdiger gave a talk at a workshop I hosted last year (and I should finish editing his video soon in case you are interested), but I don’t recall anything he said as suggesting there truly isn’t an objective reality. Again, maybe I’ll pester him about it at some point. At any rate, that leaves Chris. I know that Chris has had to defend QBism against charges that it is solipsistic. He swears it isn’t. However, in my mind, a truly solipsistic argument is the only legitimate counter to claims of an objective reality since it can’t really be refuted, e.g. if you told me that I existed solely in your imagination, I would have absolutely no way to disprove that to you (even your death wouldn’t disprove it to you since you would be dead). Even so, I don’t think QBism necessarily requires the denial of an objective reality. Either way, I plan to pester Chris about this as well.

So that, in a nutshell, is why I think people are mistaken when they claim that physics somehow denies the existence of an objective reality. Quite the opposite, I say: physics requires it.

Neal Stephenson on Leibniz, monads, and more

Posted in Uncategorized on September 9, 2014 by quantummoxie

I have finally gotten around to editing the videos from the workshop I ran with Dean Rickles last March. The first few have been posted on the workshop website and I’m going to slowly repost them here as well (in lieu of writing another blog post — trust me, Neal Stephenson is far more compelling than anything I could write). So, without further adieu, here is the estimable Neal Stephenson, recorded at Trinity College, Cambridge by yours truly back in March. Stay tuned for more.

CPT-symmetry and the nature of time

Posted in Uncategorized on June 18, 2014 by quantummoxie

Our follow-up to last summer’s PRL outlining a quantum resource theory for CPT-symmetry has hit the arXiv and been accepted for publication (without mods!) in PRA. We’ve got some further generalizations we’re starting to work on, but one of the things this work has crystallized in both my mind and many other people’s minds is that true “time-reversal” is really CPT-reversal. Nevertheless, there are still some pesky questions about time that persist, despite Ken Wharton’s argument that there’s really no funny business going on at all. Ken has tried to convince me to buy into the block universe explanation. I’m still not entirely sold on the idea, but I have come to believe that the problem of the nature of time as an “isolated” problem is less important than the relative nature time to space. In other words, I think the more important question that needs to be addressed is, why does the metric tensor that describes the universe have at least one negative eigenvalue, i.e. why is the sign of the time component always opposite to the sign of the spatial components in the metric?

Ken might answer that this is an artifact of our perception. For example, I might say that “normal” geometry, i.e. the Euclidean geometry of everyday life, doesn’t exhibit this feature. Ken might counter that that’s just a result of the fact that we perceive one of the four dimensions differently even though they’re all really the same. But that still leaves the question as to why we perceive that one dimension differently. It clearly is independent of the human mind since other species “perceive” time and time does appear to have some kind of preferred direction while space does not. Either way, the fact of the matter is that the metric tensor that describes the universe that we observe and measure has a negative eigenvalue, regardless of whether the space is flat or curved. We can’t magically force the metric to have only positive eigenvalues. Science is about describing what we can reliably measure with a healthy dose of Occam’s Razor thrown in for good measure. The simplest description of the universe’s geometry that matches experiment forces the presence of at least one negative eigenvalue in the metric tensor. Why? That’s the question that needs to be answered.


Get every new post delivered to your Inbox.

Join 431 other followers