Table of contents
- After 40 years, string theory hasn't delivered; it's time to rethink the foundations of physics.
- Despite its simplicity, the Standard Model of particle physics perfectly matches all experimental results, yet it leaves us questioning why the universe is built this way.
- Why does the universe have such a specific pattern of particles and forces?
- The world is full of claimed solutions to quantum gravity, but none are truly satisfactory or testable.
- Protons don't decay, despite all predictions and experiments.
- Supersymmetry is a beautiful idea that predicts every particle has an unseen super partner, but so far, there's zero evidence for them.
- Exploring extra dimensions in physics seemed promising, but we've never found any evidence beyond our familiar four.
- String theory promised to explain everything, but after 40 years, there's still no evidence it works.
- After 50 years, it's time to admit some ideas just don't work and move on.
- String theory isn't working out, and it's time to explore new ideas in physics.
- Sometimes the impossible becomes possible when you rethink the fundamentals.
- Understanding spinors in Euclidean vs. Minkowski space reveals a new unification of internal and space-time symmetry.
- Quantum field theory's two main formalisms can't be analytically continued between real and imaginary time.
- Imaginary time transforms complex quantum theories into statistical mechanics, revealing hidden symmetries and new perspectives.
- The Dirac operator isn't a scalar; it's a vector under Lorentz transformations.
- Space-time is right-handed, not left-right symmetric.
- Exploring the asymmetric nature of twister geometry and its implications for general relativity is an ongoing journey of discovery.
After 40 years, string theory hasn't delivered; it's time to rethink the foundations of physics.
The reason we don't see any extra dimensions is that there aren't any. Most serious people in the subject have stopped working on the ideas that these things are failures. It's all about four dimensions. What if our quest for unification in physics has been fundamentally misguided? Since the 1980s, string theory promised a unified framework, but after 40 years, it's failed to deliver. Now, a growing number of physicists are calling for a radical rethinking of our foundations.
Enter Peter Woit from Columbia University, who earned his master's from Harvard and his PhD in particle physics from Princeton. Known for his incisive writings on Not Even Wrong, his textbook Quantum Theory Groups and Representations, and a fresh approach to a theory of everything, Woit isn't just pointing out flaws in mainstream fundamental physics; he's proposing something disruptively new. In this episode, we'll dive into the Standard Model, explore the problems with supersymmetry, and uncover why Woit believes that the solution to unification lies in understanding imaginary time. At the core of his approach are spinors, which researchers like Roger Penrose and Michael Atiyah call the most mysterious objects in the world.
Welcome, Professor Peter Woit. It's an honor to have you back on the podcast again. It's your second round, I believe. "Yes, that's right. Thanks. Thanks. I'm glad to be back." Today, you have a talk prepared for this conference called Rethinking the Foundations of Physics and What is Unification is the theme of this year. So, take it away. "Okay. So, and we'll see. I mean, this is fairly sketchy. I'll have to make some excuses for the, to really go into a lot of the things I'd like to go into would take quite a while. But I thought this is what I could do that I think I could try to convey it in a relatively reasonable amount of time. So, let's just start with that."
"So, what I wanted to do is first go over, you know, what it is, at least to me, what unification is, what are the things that we're trying to unify. And then explain kind of what the kind of current paradigm for what this kind of unification might look like that we've been kind of living with for the last basically 50 years. And then, so I want to explain what that is. And then I want to say just a little bit about what I've been trying to do and what's gotten me very excited in the last few years, which is what, to me, I believe is kind of a quite new idea about, you know, about how to do unification or about how to do a substantial part of unification in a new way, which doesn't have the same kind of problems as the things that we've been living with for the last 50 years. So, that's just the outline."
"Okay. So, to start, so, I mean, this is, you never know kind of how much to try to tell people about this. It's just kind of, there's a standard outline of what's the standard model. But, you know, we have this incredibly successful theory called the standard model. And it has, it's basically a fairly simple conceptually object, you know, once you get used to certain kind of technical ideas about the mathematics and the physics. And it basically says that, you know, there are three forces in the world and they're due to these U1, SU2 and SU3 gauge fields. There's the, basically the electromagnetic, the weak and the strong force. And then the matter, you know, is spin one half fermions and there's some specific pattern of charges, which are the, you know, the couplings to these three different kinds of forces. And I won't write out, there's kind of a standard table of these. It's kind of an intriguing pattern we don't quite understand, but it's a pretty simple pattern. So that's forces, that's matter. And then the one probably most mysterious part of it is the Higgs field. And so this Higgs field is this spacetime scalar field, which breaks the U1 and SU2 down to a U1 subgroup. And that gives masses to the weak, to the SU2 gauge bosons and to the matter. So that's pretty much all there is. And so if somebody just tells you that, knowing the basics of the geometry and how this is supposed to work, you could reconstruct the whole theory. You can reconstruct the whole theory once I tell you the charges. And then there's going to be a lot of undetermined parameters in the thing."
"Okay, so history. So this basically, we're kind of a bit over 50 years out from this. Okay, so for here, just a quick clarification. See how it has U1 cross SU2, but then it goes down to just U1. And some people may be wondering, okay, you had electroweak unification, but you still have electromagnetism plus the weak force. Where did the weak force go? You're saying, correct me if I'm incorrect, that U1 is the only unbroken symmetry left after the Higgs mechanism. Okay, understood. Okay, so the history. Yeah, so this is pretty much, there's a long history of this, but it kind of came together pretty quickly in a few years. And, you know, in April 1973, you know, you could write down this theory and people started to realize, you know, what they had. And it took them a while to really, you know, to gather the experimental evidence to be convinced that this was really the right thing. But it was there in April of 73."
"Okay, and so now, I mean, the most amazing and bizarre aspect of this whole situation is that, you know, this relatively simple theory, there are basically all experimental results agree exactly with it. There's no such thing as some, you know, interesting experimental result, which you can't, you know, explain with this theory. And there's some technicalities about, you know, the first version of this story didn't have masses for the neutrinos, but it turns out you can throw in some right-handed neutrino fields, and it all works exactly, you know, as you expect so far. There isn't any data. The only kind of data that people talk about that we're not sure what to do about is often more kind of astrophysical data, things like dark matter and dark energy and questions about cosmology.
Despite its simplicity, the Standard Model of particle physics perfectly matches all experimental results, yet it leaves us questioning why the universe is built this way.
In April of 1973, scientists began to realize the significance of their findings. It took them some time to gather enough experimental evidence to be convinced that their theory was correct. However, once they did, they observed that all experimental results agreed exactly with this relatively simple theory. There were no experimental results that couldn't be explained by it. Initially, the theory didn't account for neutrino masses, but by incorporating some right-handed neutrino fields, everything worked as expected. The only data that remains uncertain often pertains to astrophysical phenomena, such as dark matter, dark energy, and cosmological questions. Every experiment conducted at short-distance scales has consistently agreed with this theory.
This situation presents a problem of unification. Historically, experimental results have often disagreed with prevailing theories, providing clues on what adjustments were necessary. However, in this case, there are no such discrepancies. On the other hand, general relativity describes space-time as a three-plus-one-dimensional pseudo-Riemannian manifold, where one direction has a negative metric. Locally, it resembles Minkowski space-time, and the gravitational force is described by the curvature of this space-time, governed by the Einstein-Hilbert action or the Einstein equations. This is a classical theory, unlike the quantum nature of the standard model. Despite being over a hundred years old, general relativity has not required changes and agrees precisely with all gravitational force measurements.
The problem lies in the fact that both theories are geometrical and based on fundamental symmetries, yet they leave some questions unanswered. There is no evidence suggesting anything wrong with either theory, but they don't explain everything. For instance, why SU1, SU2, and SU3? Why these three gauge groups and forces? Each of these groups has a free parameter, a coupling constant describing the force's strength, resulting in three numbers. One of these is the strength of the electromagnetic force, but why these specific numbers? A better theory would ideally explain the values or ratios of these numbers.
For those unfamiliar with the technicalities, it might seem arbitrary, like saying the universe is composed of a circle, triangle, and square. U1 is essentially a circle in the complex plane, SU2 can be seen as two-by-two unitary matrices with determinant one or a three-dimensional sphere, and SU3 consists of three-by-three unitary matrices with determinant one. But why these groups? They are among the simplest possibilities, but the question remains: why not something else?
A quick retort might be that no matter the groups, whether E8 or G2, the question of "why these groups?" would still arise. This leads to the broader problem of unification. Additionally, there are questions about the matter particles: why are they spin one-half fermions? Why do they have this specific pattern of charges? These charges are integers that describe how they couple to U1, SU2, and SU3. Moreover, why do they come in three generations, repeating the same pattern? This discrete structure is small and manageable, but its origin remains a mystery.
Why does the universe have such a specific pattern of particles and forces?
Let's discuss some key points related to grand unification and the associated problems. One question that arises is why matter particles are spin one half and why they are fermions. Additionally, why do they have a specific pattern of charges? This pattern involves integers that indicate how they couple to the U1, SU2, and SU3. Furthermore, these particles come in three generations, repeating the same pattern three times. This discrete structure seems small and manageable, but it suggests there should be an underlying explanation.
Another significant topic is the Higgs field, a scalar field with potential energy chosen to have a minimum away from zero, thus breaking symmetry. Questions arise about the origin of this potential energy function and why the Higgs field is a complex doublet that transforms under SU2. The matter fields get their masses from the strength of their coupling to the Higgs field, known as Yukawa couplings. Each different matter field couples to the Higgs field with a different parameter, raising questions about the origin of these parameters.
There are also technical problems with the quantum field theory of the standard model. Computations are often done in perturbation theory using Feynman diagrams, which work well for small couplings but not for larger ones. For SU3, lattice gauge theory provides a way to define the theory non-perturbatively, but incorporating matter particles complicates things. Specifically, for chiral gauge theories like SU2, which couple differently to left and right-handed spinors, there is no known way to discretize and define the theory non-perturbatively.
A more widely discussed issue is the quantization of general relativity. While general relativity works well as a classical theory, quantizing it using standard methods leads to renormalizability problems and infinities that cannot be handled traditionally. There is no completely consistent, non-perturbative definition of quantum gravity, either standalone or coupled with the standard model. However, many proposals exist, such as string theory and loop quantum gravity, each claiming to address the problems of general relativity. Depending on one's perspective, one might believe there is no solution or an abundance of potential solutions, particularly in the context of string theory, which suggests an exponentially large number of possible solutions depending on string vacua.
The world is full of claimed solutions to quantum gravity, but none are truly satisfactory or testable.
One way of saying it is that there are plenty of people who claim that they have an idea, here's the idea, here's a way to solve quantum gravity, whether it's string theory or loop quantum gravity, or a hundred other proposals. Many people claim to have at least a plausibility argument that they've got a way to handle the problems of general relativity. Depending on how seriously you take these claims, you could say either there is no such thing, or there's actually a huge number of them. If you believe everything that a lot of the string theorists would like to be true, they would like to say that string theory gives you such a thing, but it may give you an exponentially large number of such things, depending upon these questions about string vacua, et cetera. So, there are maybe two ways to say the problem: one is that this problem has no solution at all, and the other is to say that the world is full of claimed solutions, but none of them really seem to actually explain very much, have any way to test them, or are satisfactory.
Now, I want to start on what has been happening since April 1973, when it became clear what these problems were. The first thing that happened is a few months later, Howard Georgi and Shelly Glashow came up with what's called the first example of a grand unified theory. They were addressing the problem of the three groups with three constants, trying to improve the situation by fitting them together as subgroups of one larger group, typically SU5 or SO10. Instead of having three coupling constants, you have one coupling constant. This results in relations between the three coupling constants.
If you write down the theory for the bigger Lie group, it has just one coupling constant. You then have to explain why we see three coupling constants. The problem is that if there was just an SU5 theory, there would be one number that determined everything. But we see three things and three numbers, so you have to explain why we see three things and how you go from one number to three numbers. This involves setting up a new kind of Higgs mechanism. Above a certain energy scale, the so-called GUT energy scale (around 10^15 GeV), you see the full SU5 theory with one coupling constant. Introducing symmetry breaking at this scale, you evolve down to lower energies, where the U1, SU2, and SU3 couplings evolve differently.
You often see a graph of these three coupling constants coming together at a point where they unify to SU5. The next step is to address matter. You need to explain how the irreducible representations of U1, SU2, and SU3 that all your matter fields fit into transform into those symmetries. This involves ensuring that the charges from the subgroups fit together into one representation of the bigger group. For SO10, all known particles fit into one nice representation, the spinor representation. However, you also need to introduce new Higgs to explain why we don't see the big group of symmetries and instead see the smaller group of symmetries.
Protons don't decay, despite all predictions and experiments.
In the process of understanding particle physics, you have to explain how all those things, those numbers you get from the subgroups, fit together into one thing. This involves showing how all those matter things fit together into a representation of the bigger group. Essentially, you have a generalized notion of a charge for SU5, and you have to pick the SU5 charge of your basic particles. Then, you must ensure that when you examine the U1, SU2, and SU3 subgroups, it gives you the correct list of charges that we know about. This is a technical task, but it can be done quite nicely, especially for SO10. All the known particles fit together into one nice representation of SO10, the spinor representation. However, you also have to introduce new Higgs to explain why we don't see this big group of symmetries and why we see the smaller group of symmetries. Just as SU2 cross U1 breaks down at the energy where the vacuum is only invariant under U1, the vacuum cannot be invariant under SU5 or SO10. Therefore, you need to introduce some more dynamics to break it down to this, and later break it down again to U1.
Initially, George and Glashow got very excited about this because it not only provided a pretty pattern and explanation for some numbers but also offered new predictions of new physics. Specifically, by combining SU2 with the weak force and SU3 with the strong force, quarks can decay into leptons. This means protons, in particular, are not stable; the quarks inside a proton or neutron will eventually decay into another quark and a couple of leptons. There was a nice calculation of how fast this should happen, and the initial numbers suggested it should occur very slowly, consistent with the fact that we don't observe proton decay. Consequently, people started conducting experiments to look for proton decay at the predicted rates. However, the problem has been that protons don't decay. Despite building bigger detectors and looking more carefully, there has been no evidence of proton decay. The initial SU5 and SO10 theories had rough estimates of the proton decay rate, but the actual bounds are now way above those predictions, proving them wrong.
Interestingly, Georgi and Glashow eventually gave up on this idea. Unlike others who persist with such theories, they acknowledged the failure and stopped working on it. If you talk to them today, they'll admit it was an exciting idea that didn't work out. Georgi, for instance, is now working on un-particles. They moved on about 10 to 15 years after the experimental results came in, conceding that it was a bad idea.
However, a more disturbing situation is that many basic textbooks still tell this story to graduate students, presenting it as a wonderful idea about unification without clearly mentioning that it doesn't work. Supersymmetry was another part of our standard paradigm. In the earliest standard models written down in April, by December, people were already writing down supersymmetric extensions of the standard model. This can get quite technical, but the basic idea involves understanding the crucial relation between spinors and vectors. Spinors, in some sense, are a square root of vectors. They are mathematical objects that, when you take the tensor product of two of them, you get a vector. Vectors correspond to translations, and the world locally looks like a certain vector space of four dimensions, allowing translations in any four directions, corresponding to momentum or energy operators, and rotations. Supersymmetry suggests extending the standard story about momentum and angular momentum and how it all fits together.
Supersymmetry is a beautiful idea that predicts every particle has an unseen super partner, but so far, there's zero evidence for them.
In April, models were written down, and by December, people were writing down these supersymmetric extensions of the standard model. Let me explain what those are. This can get quite technical, but one way of understanding the basic idea is to recognize the crucial relation between spinors and vectors. Spinors, in some sense, are a square root of vectors. They are mathematical objects such that if you take the tensor product of two of them, you get a vector. Vectors correspond to translations, and we know that the world locally looks like a certain vector space of four dimensions. You can translate in any of these four directions, resulting in corresponding momentum or energy operators. Additionally, there is rotation.
Supersymmetry suggests extending the standard story about momentum and angular momentum, which fits together into the Poincarelli algebra, by adding new generators corresponding to the spinor direction. These new generators are anti-commuting, unlike the usual ones, and the tensor product of two spinors being a vector corresponds to the anti-commutator of two of these operators being a translation operator. This is the basic idea of supersymmetry, a beautiful concept. Starting in 1974, people took the standard model and added fields to it, allowing the definition of this extended symmetry and the new spinor generators, the Qs, or supersymmetry generators. This could also be applied to grand unified theories, turning them into supersymmetric grand unified theories.
There was a lot of enthusiasm about this idea, driven by its beauty. When you try to implement it, you find that these Qs commute with all the U1, SU2, and SU3 charges. A Q will take any particle with certain charges and turn it into a super partner, producing a different kind of particle with the same standard model charges but with spin differing by a half due to its spinor nature. Ideally, you would find two particles in the standard model related by one of these supersymmetry generators, differing by spin half and having the same charges, identifying them as super partners. However, the problem is that this beautiful new symmetry doesn't relate any two known particles; it relates everything known to something never seen before. Technically, this symmetry acts trivially on everything known.
This leads to the prediction that every particle known has a super partner, suggesting that only half the particles in the world have been seen. Some people enthusiastically embraced the idea of new particles, while others, including myself, found it implausible that such a new symmetry exists without observable effects. The Large Hadron Collider (LHC) has placed very strong limits on this, finding no super partners and zero evidence for any of this.
Another part of the unification paradigm includes supergravity and Kaluza-Klein theories. Developed a few years after the standard model, supergravity turns supersymmetry into a gauge theory, extending general relativity. The Gravitino is a partner to the Graviton, and this theory, when quantized, seems to have fewer renormalizability problems. Additionally, going back to the early days of general relativity, people explored the idea of more than four space-time dimensions to explain the U1, SU2, and SU3 charges through internal dimensions. This idea, although long considered wrong, became a significant part of the paradigm people were examining.
A quick question arises: Supersymmetry can be formulated at the classical level, correct? Yes, it can. If you put supersymmetry on general relativity, then you have a Gravitino. However, you don't have a Graviton at the classical level; the Gravitino appears in the quantum version.
Exploring extra dimensions in physics seemed promising, but we've never found any evidence beyond our familiar four.
We had the super gravity theories. Going back to the early days of general relativity, people had been looking at what happens if you have more than four space-time dimensions. One thing you might try to do is explain where the U1, SU2, SU3 come from by postulating more than four space-time dimensions. It's these other so-called internal dimensions which explain everything. This idea was wrong for a long time, but it became a big part of the paradigm that people were looking at.
A quick question arises: Can supersymmetry be formulated at the classical level? The answer is yes. If you're putting supersymmetry on General Relativity (GR), then you have a Gravitino. However, you don't have a Graviton at the classical level. The Gravitino appears in the quantum version. The classical version of supersymmetric general relativity does have properties that are studied, but the problem with all supersymmetric theories is that you extend your standard variables with anti-commuting variables to get Fermions. Classically, it's a weird subject because you have non-commuting classical variables, which means it doesn't correspond to our intuitions about classical physics.
Someone like Alain Konis might be comfortable with classical non-commutativity, but he is more interested in a specific sort of non-commutativity, sometimes called Z2 graded commutativity or super commutativity. In this context, things don't commute, but the extent to which they don't commute is minor, like picking up minus signs when interchanged. Non-commutative geometry, as discussed by people like Alain Kon, generally means something more seriously non-commutative. Some mathematicians call this super commutative, and in standard commutative geometry, algebraic gadgets that square to zero and anti-commute are part of the story.
This discussion brings us to the period when I was an undergraduate starting in 1975, taking quantum field theory courses from 1976 to 1977, and paying attention to what was going on. At that time, people were talking about these ideas as the answer to unification problems. For example, Hawking's initial lecture for his professorship titled "Is the End in Sight for Theoretical Physics?" suggested that super gravity in the Kaluza-Klein version might give us a quantum theory where everything fits and explains everything. However, none of this worked out because we've never seen any extra dimensions or anything besides four dimensions. The Kaluza-Klein idea never led to anything verifiable.
In the early 80s, people were also studying string theories. The first superstring theory paper suggesting that it could describe gravity was published just a month after the standard model was in place. This idea exploded in 1984 when Witten got involved, leading to serious interest in unification through string theory. The basic idea was to think of the basic objects of the theory as one-dimensional extended objects instead of point particles and fields based on those particles. The superstring theories aimed to bring together everything: an E8 GUT, super gravity as a low energy limit, and extra dimensions of Kaluza-Klein.
String theory promised to explain everything, but after 40 years, there's still no evidence it works.
The study of string theories has a long history. Although we don't delve deeply into it here, it's interesting to note that the first superstring theory, which proposed that gravity could be described using a superstring, was published just a month after the standard model was established. This concept gained significant traction in 1984 when Witten entered the field, sparking serious interest in unification through this approach. The basic idea shifted from thinking about point particles and fields to considering one-dimensional extended objects as the fundamental components of the theory.
Superstring theories aimed to integrate various concepts, such as an E8 GUT, supergravity as a low-energy limit, and extra dimensions of Kaluza-Klein theory. This comprehensive approach appealed to many because it promised to unify numerous previously unworkable ideas into a single, all-encompassing theory. People believed they had found a theory of everything, with Witten, a recognized genius, enthusiastically promoting it as the future of physics.
However, 40 years later, there is no experimental evidence supporting any components of this theory, including the strings themselves. Despite extensive study, no observations have indicated any connection to these theoretical constructs. Physicists working in this area often fail to clearly communicate the extent to which these theories have not worked out. They embed successful, well-established concepts into larger, more complex structures, such as larger gauge groups or additional dimensions, hoping to uncover new symmetries or computational advantages. Yet, there is no evidence for any components of these new structures.
The challenge arises when these larger structures, despite their theoretical elegance and symmetries, fail to manifest in observable phenomena. Physicists then have to explain why these features are not seen, often by proposing mechanisms to make these dimensions or superpartners undetectable. This process of adding layers of complexity to account for the lack of evidence turns the initially elegant theories into something increasingly convoluted and detached from reality.
This pattern of theoretical failure is common: a promising new idea is proposed, but when experiments fail to confirm its predictions, theorists either abandon it or complicate it further to avoid admitting it was wrong. Unlike George and Glashow, who accepted their mistakes and moved on, many in the field continue to modify their theories, making them more complex and less predictive. This has led to 50 years of increasingly ugly and unpredictive theories, with few willing to acknowledge their failure.
Many serious researchers have stopped working on these theories, not necessarily declaring them failures but simply seeing no way to advance them. They might still consider these ideas beautiful but recognize their limitations. For instance, if asked, Witten might still appreciate the elegance of these theories despite their lack of empirical support.
After 50 years, it's time to admit some ideas just don't work and move on.
The obvious conclusion is that this was just the wrong idea. However, it is incredibly hard to get people to even admit that this is a sensible interpretation of what's happened in the last 50 years. This difficulty is why I'm going through all this. Anyway, this is more of what I wanted to say on this topic.
What's actually happened is that many people continue to push through these old ideas that don't work. However, the most serious people in the subject have kind of stopped working on these ideas. They don't openly declare these ideas as failures, but they have ceased working on them. If you ask them about it, they often say, "I just don't see how to push this any farther. I still think it's a beautiful idea." They might add that unless some experiment comes along and provides a new hint on how to make these things work, it looks kind of hopeless. As a result, they have stopped thinking about it every day.
The new ideology seems to be that instead of admitting that this idea failed, people now suggest that thinking about unification is no longer something a serious person should do because it's hopeless until someone has a brilliant new idea. Until experimentalists help us out, we won't be able to move forward with this. This sentiment is common among theorists, who now see the idea of thinking about unification as something that only a crank would do. They believe that only an amateur or crank would fail to realize that the smartest people worked for 50 years on this and thought it was the best possible way, yet they haven't been able to make it work.
The question then arises: should we consider unification attempts outside of string theory or not even consider string theory unification? String theory becomes a complicated question. One way to address this is to think of string theory, GUTs, supersymmetry, and extra dimensions as the paradigm we've had for 50 years. The problem for anyone trying to say that what has been done for 50 years doesn't work and that something completely different is needed is that it's a hard sell. People argue that geniuses have been working on this for 50 years, and these are all great ideas. How can you tell them that it's all just wrong? This is akin to crackpots who claim Einstein must be wrong.
It's always been a hard sell to say that everything done for all this time should be forgotten in favor of something quite different. It would become less of a hard sell if people would admit that this was all just wrong and that we need to look at very different things. However, you're not seeing that case being made. Instead, people are reluctant to go back to 1973 and consider different things than those started back then.
On the one hand, you put something into the oven, and it needs some cooking. There's the fear that if you take it out too soon and prematurely dismiss it, like perhaps SU5 was a great idea, you don't dismiss it after the first year; you investigate it some more. But then there is the opposite phenomenon of overcooking, and you have to admit when something has become burnt, maybe it's been burnt after 50 years in the oven.
This raises the question: at what point do you give up an idea? My argument with the string theorists from the beginning was that they really have to give up because it hasn't worked out. Their argument was that they still think it's the best thing they know how to do and worth pushing forward. This has been a hard argument to have, but over the last 20 years, it has become clearer that this stuff just doesn't work. The argument has shifted from "we want to keep working on it" to "maybe within five or ten years, we'll have something new" to now saying, "it may take 500 years for us to make any progress on this.
String theory isn't working out, and it's time to explore new ideas in physics.
My argument with the string theorists has always been that they need to give up because their approach hasn't worked out. Their counter-argument was that they still believe it's the best thing they know how to do and worth pushing forward. This was a difficult point to argue against, but over the last 20 years, it has become clearer that this stuff just doesn't work. The argument has shifted from "we'll make progress in five or ten years" to "it may take 500 years to make any progress." This is taking longer than I thought.
For those who have followed this talk so far, this is a quick recapitulation of the standard model and the state of affairs in physics. In about 40 minutes, we've covered the state of physics from 1915 to the 1970s and then to the present day. Although I haven't explained a lot about it, the bottom line is more depressing: you shouldn't actually study any of it. Instead, you should try to find something else to do.
Now, let me talk about what I've been trying to do. When I was a graduate student, I worked on lattice calculations using SU3 gauge theory, which involved discretizing and putting out a lattice. I thought this was great, but then I wondered about the matter particles and what happens when they are put on the lattice. I realized that matter particles are spin one-half and their spin geometry is very complex. It is not obvious how to capture and preserve that geometry when discretizing things. There is a long history of people trying to put spinor fields on the lattice, leading to many interesting problems. This is where I first started thinking about these issues.
I had a vague idea that maybe it should be possible to address these problems, but at some point, I gave up on it. This wasn't due to a lack of experimental evidence, but because everything I knew about the subject suggested it was implausible. I convinced myself that the way space-time symmetries work made it impossible. However, in the last three or four years, I revisited these ideas, especially after teaching courses on quantum mechanics and quantum field theory (QFT) and writing a book about it. I started to understand more precisely how these symmetries work and realized that my previous assumptions were incorrect. There is now a perfectly coherent way of thinking about what I once thought couldn't happen.
This realization came after writing my book on quantum theory and representations and teaching the course several times. Although I haven't fully written down these new ideas, the process of writing and teaching has significantly contributed to my understanding.
Sometimes the impossible becomes possible when you rethink the fundamentals.
I started to see that there was a perfectly coherent way of thinking about what I thought couldn't possibly happen. There were now perfectly good reasons to believe that it could happen. This realization occurred more after I had finished writing the book on quantum theory and representations. The process of writing that book first got me motivated to clearly document the story of these space-time symmetries. I aimed to get everything written down to understand exactly how certain phenomena happened. As I did this, I realized that what I was convinced would explain why something couldn't work wasn't there.
The idea that seems promising is centered around four dimensions. The reason we don't see any extra dimensions is that there aren't any; it's all about four dimensions. We should look very carefully at four dimensions and ask what is very special about four-dimensional geometry. There are many interesting things that happen only in four dimensions, particularly the geometry of spinors and twisters. Twisters, an idea by Roger Penrose, are very beautiful for understanding conformal geometry in four dimensions and are deeply tied to four-dimensional geometry.
One thing to note is that if you look at the literature on theories like supersymmetry, supergravity, or string theory, you find a strange technical problem. Our space-time has a Minkowski metric with a minus sign on the distance squared in time. When writing these theories, you encounter technical problems if you try to do it in this indefinite Minkowski signature. To address this, theorists assume all four dimensions are the same, write the theory in that context, and then use Wick rotation to recover what happens in Minkowski space-time. This relationship between Euclidean and Minkowski signatures is a significant topic that indicates something we don't fully understand.
The main new idea I'm proposing is that this Wick rotation changes the geometry of spinors in a fundamental way. The geometry of spinors in Euclidean signature and Minkowski signature is quite different. The idea, which I initially thought couldn't work but now believe does, involves the four-dimensional rotation group breaking up into two SU2 factors. When you Wick rotate to Minkowski space-time, one of these factors becomes a space-time symmetry.
Understanding spinors in Euclidean vs. Minkowski space reveals a new unification of internal and space-time symmetry.
There are other parts of the subject where you look at something and say, "Wait a minute, there isn't a clear explanation for exactly what's going on here." Wick rotation was a place where that happens in the standard model. The main new idea is to claim that this Wick rotation, if you think about your geometry in terms of spinors, changes the geometry of the spinors in a very fundamental way. The geometry of spinors in Euclidean signature and the geometry of spinors in Minkowski's signature is actually quite different.
The basic idea, which I had going way back and didn't think could work but am now convinced does, is that in the four-dimensional rotation group, it breaks up into two SU2 factors. When you Wick rotate to Minkowski space-time, one of those two factors becomes a space-time symmetry, and the other becomes an internal symmetry. This provides a new unification of internal and space-time symmetry. On the Euclidean side, these things get unified, involving only the degrees of freedom we know about, with nothing extra. The new thing is to realize that there's a very important subtlety when you try to make spinors go back and forth between Minkowski and Euclidean signatures.
To summarize, there's the Pythagorean theorem: A squared plus B squared equals C squared for two dimensions. In three dimensions, it's A squared plus B squared plus C squared equals the hypotenuse. In Einstein's theory, you have something plus something plus something minus something else, and that minus causes issues, such as oscillations in the Feynman path integral. Wick rotating takes that minus sign, which is technically an imaginary number, and turns it into a positive, making the space much nicer to work in.
Additionally, there's a low-dimensional coincidence with spin four being akin to SU2 cross SU2. This is more SO4, but the idea is that Wick rotation involves changing the time variable to complex, making time purely imaginary. This cancels the minus sign, making everything positive. This process is referred to as going from Minkowski (real time) to Euclidean (imaginary time). Even in the simplest quantum mechanical models, making time imaginary is the simplest version of Wick rotation.
However, in quantum field theory, this gets technical. Field operators depend on time, and making them depend on complex time means conjugating by the Hamiltonian operator in the Heisenberg picture. If you try to go to imaginary time, you conjugate by the exponential of the imaginary time times the Hamiltonian. The Hamiltonian's eigenvalues are the energy, which can be arbitrarily high positive energy. This creates a problem: with positive tau, one operator makes sense, while the other becomes exponentially large, and vice versa for negative tau. This fundamental issue affects everything we know about quantum field theories.
Quantum field theory's two main formalisms can't be analytically continued between real and imaginary time.
The operator in question is the exponential of the imaginary time, times the Hamiltonian. However, the Hamiltonian's eigenvalues are the energy. This means it has a spectrum of positive energy that extends to infinity in the cases of interest. For a typical theory, even a simple particle must have positive energy, but it can also have arbitrarily high positive energy.
The issue arises with the two operators, ( e^{\tau H} ) and ( e^{-\tau H} ). If (\tau) is positive, ( e^{-\tau H} ) makes sense because it is ( e ) to the minus something positive times something positive. Conversely, ( e^{\tau H} ) becomes exponentially large. If (\tau) is negative, the situation reverses. This fundamental issue means that you can't analytically continue the theory; you can't make time complex and have it behave as desired. The rules for what happens to the field just don't make sense in this context.
In the operator formalism, this issue is evident. However, in the path integral formalism, the opposite behavior is observed. When written as path integrals in imaginary time (Euclidean space-time), the path integrals are ( e ) to the minus something positive and large, which makes perfect sense. This results in integrating some kind of Gaussian thing or something that falls off nicely at infinity. In contrast, in Minkowski space-time or real time, you end up integrating over an infinite-dimensional space ( e^{i \times \text{something}} ), which involves integrating a wildly varying phase over an infinite-dimensional space. This doesn't make sense as a measure or a real integral.
The two formalisms used in quantum field theory have opposite behaviors. While people talk about using them to go between imaginary and real time, it isn't feasible. One works well in real time as a formal object, and the other works in imaginary time. There is no theory that allows for a seamless transition between the two.
Wick rotation, defined in the Feynman case, doesn't apply to the operator formalism. The two main formalisms for writing down a quantum field theory have one working in one case and not in the other, and vice versa. There is no full theory or formalism that depends on complex time analytically and allows for analytic continuation between time and imaginary time.
This problem exists in both directions: starting with Euclidean and trying to get to Minkowski, or vice versa. Only one formalism works depending on where you start, and you can't transition from one to the other.
However, there is something that can be done. While you can't analytically continue the theory, operators, states, measures, and all these things, you can define Wightman functions. These are vacuum expectation values of operators. By taking a product of two operators at two different spacetime points, applying them to the vacuum, and taking the inner product with the vacuum again, you get functions dependent on X and Y. These carry most of the information about the theory.
In real time and an operator formalism, you can compute these objects and characterize the theory by them. These objects are distributions, not functions, and are more like delta functions. In imaginary time and the path integral formalism, similar things can be taken, which correspond one-to-one with the Wightman functions, except they are symmetric. This results in a very different kind of theory, with no states or operators, just measures and integrals.
Imaginary time transforms complex quantum theories into statistical mechanics, revealing hidden symmetries and new perspectives.
Interchanging X and Y results in something different. Technically, these are distributions, not functions. They resemble delta functions and don't make sense as actual functions, but you can take their convolution with functions to get something meaningful. This is feasible in real-time and operator formalism. In imaginary time and path integral formalism, you can take similar things, which are moments of these measures. These path integrals correspond in a one-to-one way with the whitening things, except they are symmetric.
The calculation and theoretical setup are very different. There are no states or operators, just measures and integrals, resembling statistical mechanics. One amazing aspect is that if you take your imaginary time to have a finite extent of size beta and perform the calculation, it becomes a statistical mechanical calculation at a temperature given by beta equals one over K times the time. Thus, the path integral formalism is much more like a statistical mechanical system, very different from the operator formalism. The output includes Schwinger functions, which can be analytically continued to the Whiteman functions.
Now, addressing SO4 and its break to Lorentz, the theory should be considered in Euclidean space-time or imaginary time to compute the Schwinger functions. To have states and operators in the operator formalism, you need to reconstruct the real-time theory from the imaginary time theory using the Osterwalder-Schrader reflection. This involves picking one direction as the imaginary time and having an operator reflect you in that direction.
In real-time, there is no distinguished direction of time, with positive and negative time-like cones. However, in Euclidean space-time and imaginary time, you must break the SO4 symmetry and pick a distinguished direction. This introduces another problem of time, distinct from the general relativity (GR) versus quantum mechanics (QM) problem. The realization that Euclidean theory has no operators or states and requires breaking the SO4 symmetry to pick an imaginary time direction was a significant breakthrough.
This issue is well-documented for scalar field theories but remains mysterious for spinors. My proposal is that a space-time symmetry in the Euclidean QFT becomes an internal symmetry in the Minkowski QFT due to the reconstruction procedure and the introduction of the ultraviolet refraction operator with spinors. Spinors are fundamentally different in Minkowski and Euclidean space-time. In Euclidean space-time, the rotation group SO4 has a double cover, two copies of SU2 (left and right), with matter particles as vial spinors, either left-handed or right-handed. The standard story is that vectors are the tensor product of left-handed and right-handed ones.
The Dirac operator isn't a scalar; it's a vector under Lorentz transformations.
Because of what you have to do when you try and perform this reconstruction procedure, you introduce this ultraviolet refraction operator when you do it with spinors. That's the basic, one basic thing I'm saying now.
Here's just a couple of minutes on spinors before I proceed. One reason this is important is that spinors are really different in Minkowski and Euclidean space-time. In Euclidean space-time, the rotation group SO4 has this double cover, which is two copies of SU2, referred to as left and right. The matter particles are these vial spinors that are either the left-handed one or the right-handed one, represented by SU2 acting on C2. The standard story about Euclidean space-time is that if you want vectors, you take the tensor product of the left-handed ones and the right-handed ones.
In Minkowski space-time, you've got spin 3-1, which involves a different treatment of one direction. This is a very different group; it's not SU2 cross SU2, but SL2C, which consists of two by two complex invertible matrices with determinant one. There is only one kind of spinor in some sense, acting on a C2. You have one kind of spinor, which I'll call S, but you can also look at the complex conjugate and the complex conjugate B. The complex conjugate is a somewhat different thing, unlike SU2. In Minkowski space-time, vectors are tensor products of two kinds of spinors, specifically the vial spinors times their conjugates. These are two completely different things.
Now, an important thing to explain, which took me a while to realize, is about the Dirac operator. The Dirac operator really is a vector. When you write down the Dirac operator, people use upper and lower indices, making it look like a scalar. However, the Dirac operator is not a Lorentz scalar; it is not Lorentz invariant. The Dirac operator transforms like a vector under Lorentz transformations.
People don't say something directly wrong, but if you look at any physics book explaining relativistic quantum mechanics and the Dirac operator, you'll see a non-trivial transformation formula. The meaning of that transformation is very simple: the Dirac operator is a vector. Understanding the relationship between vectors and spinors clarifies this.
Finally, regarding Wick rotation, if you think about it as analytic continuation from Minkowski to Euclidean space-time, the standard way is to consider complex space-time, making all of space and time complex. The rotation group or spin group in four complex dimensions breaks up into two SL2Cs, and these complex four vectors are just a product of a spin representation of one SL2C and a spin representation of the other SL2C.
The standard story is that this is supposed to be a holomorphic or analytic story, with everything depending on analytic and holomorphic complex variables. Wick rotation is then this analytic continuation in this complex space-time. The new story I'm trying to tell is that if I'm going to do Wick rotation, I won't do it by this analytic continuation because it doesn't work or achieve the desired outcome. Instead, I will do Wick rotation starting with the Euclidean story and reconstructing the real-time theory, requiring an appropriate Osterwalder-Schrader reflection for spinor fields. I'm in the process of writing this down carefully, but what I can see happening is that when you do this, the new thing you have in your...
Space-time is right-handed, not left-right symmetric.
Now that the standard story is that this is all supposed to be a holomorphic or analytic story, everything is supposed to depend on analytic properties, and all your complex variables are holomorphic. Rick rotation is then this analytic continuation in this complex space-time. The new story I'm trying to tell is that if I'm going to do Rick rotation, I'm not going to do it by this analytic continuation because that actually doesn't work or doesn't do what I want to do. Instead, I am going to do Rick rotation starting with the Euclidean story and reconstructing the real-time theory.
I need an appropriate Osterwalder-Schrader reflection for spinor fields. I'm in the middle of trying to get this written down carefully, but what I can see happening is that when you do this, the new thing you have in your Euclidean space-time is a distinguished time, an imaginary time direction. This means you're going to have a distinguished Clifford algebra element, gamma zero. You get distinguished elements or gamma matrices, in the physicist's language, corresponding to the different directions. There is a distinguished gamma matrix corresponding to the imaginary time direction, which interchanges left and right. If you hit a left-handed spinor with it, it gives you a right-handed spinor and vice versa because it's a space-time vector.
In Minkowski space-time, what gets Wick rotated is not the tensor product of left and right-handed spinors in Euclidean space, which is the vector space-time space, but something where you've hit one of them with a gamma zero. Vectors in Minkowski space-time should be thought of as tensor products of two right-handed spinors. The geometry in Minkowski space-time is not what you thought it was; it's not the analytic continuation you thought it was. It's something different that looks spinorial, like making an analogy back to the beginning where you said that spinors can be thought of as the square root of vectors.
All these statements about vectors being different tensor products or different kinds of spinors go into discussions of supersymmetry. I'm doing something a bit different. If you look at the literature of supersymmetry and ask what happens to supersymmetry under Wick rotation, you'll find very confusing literature. This is just to explain my slogan. The last paper I wrote was a short paper trying to emphasize this from a different point of view. The slogan is that space-time is right-handed. In Euclidean space-time, you've got vectors that are tensor products of left and right, but when you do Wick rotation, you just have right times right. These left-handed spinors are really an internal symmetry. Once you've Wick rotated, they're not space-time symmetries anymore.
The slogan is that as far as space-time symmetries are concerned, you're just dealing with right-handed spinors. The left-handed spinors that you had before you Wick rotated have nothing to do with space-time; they have to do with the internal SU2 symmetry of the weak interactions.
Was there an element of chance in your theory or in your mind that made space-time right-handed versus left-handed? Oh, no, that's just a matter of convention. What I call left and right is a matter of convention. One interesting thing to say about this, and one reason for thinking about twisters, is that the standard formalism in the QFT books, where you have gamma matrices, is kind of left-right symmetric and set up to work nicely with parity-invariant theories. You have to add some things into that formalism to project out whenever you have symmetry.
Twister geometry is very asymmetric. When you write down twisters, you say that points in space-time are spinors, but they're just the right-handed spinors. Twister geometry also has this left-right asymmetry, and you have to take one of them as a fundamental thing that's telling you what the points are. I'm doing something different than the usual twister story because I'm treating vectors differently.
My next two questions may be related. Where is gravity in this? We've been dealing with flat space-time, so that's one question. The second question is what happened to Euclidean twister unification? Is that related to this?
Exploring the asymmetric nature of twister geometry and its implications for general relativity is an ongoing journey of discovery.
Twisters are a different part of the story, but the twister geometry is very much asymmetric. When you write down twisters, you say that points in space-time are basically spinors, but they're spinors of one kind—specifically, the right-handed spinors. Twister geometry also has this interesting aspect of left-right asymmetry, where you have to take one of them as a fundamental thing that tells you what the points are. However, I'm doing something different than the usual twister story because I'm treating vectors differently.
My next two questions may be related. First, where is gravity in this? We have space-time, but we've been dealing with flat space-time. Second, what happened to Euclidean twister unification? Is that related to this? To answer the first question, we know how to write down general relativity as a kind of gauge theory of formalism, specifically using SU2. Gravity, written in terms of Ashtakar variables, is done in a very asymmetric way. Essentially, I had these two SU2s: SU2 left, which is an internal symmetry related to the theory of weak interactions, and SU2 right, which is a space-time symmetry. Gauging SU2 right is how you get general relativity, but you also need to define what you're doing with the vectors. If you gauge the SU2 right symmetry and define the vectors, you can derive general relativity in Ashtakar variables.
For those interested in more details, we'll leave the links to my papers on screen. Additionally, Peter and I have a podcast on theories of everything where we delve quite in-depth into these theories. Although the concept of space-time being right-handed came a few weeks or months afterward, the Euclidean twister unification is part of this ongoing exploration. I've written various things about Euclidean twister unification, but there are many aspects I still don't fully understand. This current work is more of an answer to parts of that story, explaining things I didn't understand before.
This is an ongoing program. As I write up better versions of my past work, I start to understand things better and see them from different perspectives. This process is iterative, and sometimes I stop writing to explore new insights. The exact details are still being worked out, and while the general idea seems to come together nicely, it is not fully written up yet. If I try to formalize it, I might find that it's not quite right and will need to adjust.
Thank you, professor. We'll also link your blog on screen. Since I'm having trouble getting some of the stuff written up, I might use the blog to write about pieces of this story as I understand them. This avoids the need for a completely coherent paper but allows me to share my insights. Many people are reluctant to do this because they fear their ideas may get swiped. However, I've realized that not many people seem to understand or get interested in what I'm talking about. So, I'm not worried about people swiping my ideas. In fact, I'd be glad if anyone wants to use them and push the work forward.
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