A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

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Now, why is that? Well, many people confuse a theory of everything, as if they imagine that it's a theory in which you can compute every eventuality, and it is absolutely not that, because the computational power is very different than the question of whether or not the rules are effectively given. I've analogized it to a game of chess, and knowing all of the rules is equivalent to a theory of everything. Knowing how to play chess well is an entirely different question. But in the case of a theory of everything, or a unified field theory if you will, many people also take it to be an answer to the question, "Why is there something rather than nothing?" And I don't think that this is, in fact, what a theory of everything is meant to be either. Now, why is that? Well, because I believe at some level it is impossible for most of us to imagine an airtight argument, mathematically speaking, which coaxes out of an absolute void a something. However, there's a different question which I think might actually animate us, and which is the right question to ask of a potential candidate. And that is, "How does one get everything from almost nothing?" In the M.C. Escher drawing or lithograph, ''Hands Drawing Hands'', or ''Drawing Hands'', what we see is that the paper is presupposed. That is, if you could imagine a theory of everything, it would be like saying, "If I posit the paper, can the paper will the ink into being, such that the ink gives rise to the pens, and the pens draw the hands, which in fact manipulate the pens to use the ink?" That kind of a problem is one which is of a very different character than everything that has gone before. It is also, in my opinion, an explanation of why the physics community has been stalled for nearly 50 years since around 1973, when the standard model was intellectually in place.

Now, consider this: we have never had, in modern times, a drought where no person working in pure fundamental theory has taken a trip to Stockholm—just as a rough indicator—for contributing to the standard model. No one, in my opinion, since let's see, [https://en.wikipedia.org/wiki/Frank_Wilczek Frank Wilczek], who was born in 1951—no one born after that time has in fact contributed to the standard model in a clear and profound way. That is not to say that no work has been done, but for the most part, the current generation of physicists has, for more than 40 years and almost 50 years, remained stagnant within the standard paradigm of physics, which is positing theories that are then verified by experiment. Now my belief, which is relatively radical, is that there is no way to get to our final destination using the tools that have gotten us to where we are now. In other words, what got you here cannot get you there.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \unicode{x2215}\kern-0.7em D_A \psi = m \psi $$</div>

But the problem is that the hallmark of the Yang-Mills theory is the freedom to choose the data, the internal quantum numbers that give all the particles their personalities beyond the mass and the spin. In addition to all of that freedom is some means of taking away some of the redundancy that comes with that freedom, which is the action of the gauge group. Now we can allow the gauge group of symmetries to act on both sides of the equation, but the key problem is that if I act on connections on the right and then take the Einstein projection, this is not equal to first taking the projection and then conjugating with the gauge action. So the problem is that the projection is based on the fact that you have a relationship between the intrinsic geometry—if this is an ad-valued two-form—the two-form portion of this and the adjoint portion of this are both associated to the structure group of the tangent bundle. But the gauge rotation is only acting on one of the two factors, yet the projection is making use of both of them. So there is a fundamental incompatibility, and the claim that Einstein's theory is a gauge theory relies more on analogy than on an exact mapping between the two theories.

What about the incompatibilities between the Einstein theory of general relativity and the Dirac theory of matter? I was very struck that, if we're going to try to quantize gravity and we associate gravity with the spin-2 field \(g_{\mu \nu}\), we actually have a pretty serious problem, which is if you think about spinors, electrons, quarks, as being waves in a medium, and you think about photons as being waves in a different medium, [the] photon's medium does not depend on the existence of a metric. One-forms are defined whether or not a metric is present: it's spinors or not. So if we're going to take the spin-2 \(g_{\mu \nu}\) field to be quantum mechanical, if it blinks out and does whatever the quantum does between observations, in the case of the photon, it is saying that the waves may blink out, but the ocean need not blink out. In the case of the Dirac theory, it is the ocean, the medium, in which the waves live that becomes uncertain itself. So even if you're comfortable with the quantum, to me, this becomes a bridge too far. So the question is how do we liberate the definition? How do we get the metric out from its responsibilities? It's been assigned far too many responsibilities. It is responsible for a volume form, for differential operators, it's responsible for measurement, it's responsible for being a dynamical field, part of the field content of the system.

There are other possibilities that while each of these may be simplest in its category, they are not simplest in their interaction. For example, we know that Dirac famously took the square root of the Klein-Gordon equation to achieve the Dirac equation—he actually took two square roots, one of the differential operator, and another of the algebra on which it acts. But could we not do the same thing by re-interpreting what we saw in Donaldson theory and Chern-Simons theory, and finding that there are first-order equations that imply second-order equations that are nonlinear in the curvature? So let's imagine the following: we replace the standard model with a true second-order theory. We imagine that general relativity is replaced by a true first-order theory. And then we find that the true second-order theory admits of a square root and can be linked with the true first-order theory. This would be a program for some kind of unification of Dirac's type, but in the force sector. The question is, "Does this really make any sense? Are there any possibilities to do any such thing?"

But now, as \(\theta\) changes, the fermions are defined on the chimeric bundle, and it's the isomorphism from the chimeric bundle to the tangent bundle of the space \(U\), which is variant—which means that the fermions no longer depend on the metric. They no longer depend on the \(\theta\) connection. They are there if things go quantum mechanical, and we've achieved our objective of putting the matter fields and the spin-one fields on something of the same footing.

Well, let me just sum this up by saying: between fundamental and emergent, standard model and GR... Let's do GR. Fundamental is the metric, emergent is the connection. Here in GU, it is the connection that's fundamental and the metric that's emergent.

So we have a map into a structure group of \(U(128)\), at least in Euclidean signature—we can get to mixed signatures later. From that, we can form the associated bundle:

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Gamma^\infty(P_{U(8)} \times _{\text{Ad}}U(\unicode{x2215}\kern-0.55em S)) $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Gamma^\infty(P_{U(8)} \times _{\text{Ad}}U(\unicode{x2215}\kern-0.55em S)) = \begin{cases} \mathcal{H} \text{ gauge group}\\ \Xi \text{ sigma fields (non-linear)} \end{cases} $$</div>

So when I was thinking about this, I used to be amazed by ships in bottles, and I must confess that I never figured out what the trick was for ships in bottles. But once I saw it, I remembered thinking, 'That's really clever.' So, if you've never seen it, you have a ship, which is like a curvature tensor, and imagine that the mast is the Ricci curvature. If you just try to shove it into the bottle, you're undoubtedly going to snap the mast. So, you imagine that you've transformed your gauge fields, you've kept track of where the Ricci curvature was, you try to push it from one space, like ad-valued two-forms, into another space, like ad-valued one-forms, where connections live.

Let's think about unified content. We know that we want a space of connections \(A\) for our field theory. But we know, because we have a Levi-Civita connection, that this is going to be equal on the nose to ad-valued one-forms as a vector space. The gauge group represents on ad-valued one-forms.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ A = \Omega^1(ad) $$</div>

 

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \mathcal{H} = \Xi ⋉ \Omega^1(ad) = A $$</div>

 

Well by analogy, we've always had a problem with the Poincaré group being too intrinsically tied to rigid, flat Minkowski space. What if we wanted to do quantum field theory in some situation which was more amenable to a curved space situation?''' '''It's possible that we should be basing it around something more akin to the gauge group. And in this case, we're mimicking the construction, where \(\Xi\) here would be analogous to the Lorentz group fixing a point in Minkowski space, and ad-valued one-forms would be analogous to the four momentums we take in the semi-direct product to create the inhomogeneous Lorentz group, otherwise known as the Poincaré group, or rather its double cover to allow spin.

So we're going to call this the '''inhomogeneous gauge group''', or '''iggy'''. And this is going to be a really interesting space, because it has a couple of properties. One is it has a very interesting subgroup. Now, of course, \(H\) includes into \(G\) by just including onto the first factor, but in fact, there's a more interesting homomorphism brought to you by the Levi-Civita connection. So, this magic bean trade is going to start to enter more and more into our consciousness.

 

 

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \tau_{\mathcal{H^i}} \mathcal{H} \hookrightarrow \mathcal{G} $$</div>

And so we have a nontrivial embedding, which is in some sense diagonal between the two factors. That subgroup we are going to refer to as the '''tilted gauge group''', and now our field content, at least in the bosonic sector, is going to be a group manifold, an infinite-dimensional function space Lie group, but a group nonetheless. And we can now look at GmodHτ, and if we have any interesting representation of \(H\), we can form homogeneous vector bundles and work with induced representations. And that's what the fermions are going to be. So the fermions in our theory are going to be \(H\) modules, and the idea is that we're going to work with vector bundles of the form inhomogeneous gauge group producted over the tilted gauge group.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \epsilon = \mathcal{G} \times_{H_{\tau}} \Upsilon $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \Upsilon_H = \text{Fermi Cover} $$</div>

So, what we're talking about is something like a supersymmetric extension of the inhomogeneous gauge group analogous to supersymmetric extensions of the double cover of the inhomogeneous Lorentz or Poincaré group. Further, because this construction is at the level of groups, we've left a slot on the left-hand side on which to act. So, for example, if we want to take regular representations on the group, we can act by the group \(G\) on the left-hand side, because we're allowing the tilted gauge group to act on the right-hand side. So it's perfectly built for representation theory, and if you think back to Wigner's classification, and the concept that a particle should correspond to an irreducible representation of the inhomogeneous Lorentz group, we may be able to play the same games here, up to the issue of infinite-dimensionality.

Now, what would it mean to be able to use a gauge group in an intrinsic theory like this? We would be talking about something like an action, let's say a first-order action. And it would take the group \(G\), let's say to the real numbers, invariant, not under the full group, but under the tilted gauge subgroup.

And now the question is, do we have any such actions that are particularly nice, and could we recognize them the way Einstein did, by trying to write down not the action—and Hilbert was the first one to write that down but I, you know, I always feel defensive, because I think Einstein and Grossmann did so much more to begin the theory, and that the Lagrangian that got written down was really just an inevitability. So, just humor me for this talk, and let me call it the Einstein-Grossmann Lagrangian. Hilbert's certainly done fantastic things and has a lot of credit elsewhere, and he did do it first. But here, what we had was that Einstein thought in terms of the differential of the action, not the action itself. So, what we're looking for is equations of motion or some field \(\alpha\), where \(\alpha\) belongs to the one-forms on the group.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ dS_1 = \alpha \in \Omega^1(\mathcal{G}) = 0 $$</div>

 

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \text{ad}(P_{U(128)}) \overset{vect}{=} Cl^* = \wedge*(C) $$</div>

 

Now, this is a tremendous amount of freedom that we've just gained. Normally, we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom, and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem. I can take field content \(\epsilon\) and \(\pi\), where these are elements of the inhomogeneous gauge group. In other words, \(\epsilon\) is a gauge transformation, and \(\pi\) is a gauge potential.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ (\epsilon, \pi) \in \mathcal{G} $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon \eta = [\text{Ad}(\epsilon^{-1}, \Phi), \eta] $$</div>

So for example, I can define a '''shiab operator '''(ship in a bottle operator) that takes i-forms valued in the adjoint bundle to much higher-degree forms valued in the adjoint bundle.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon \Omega^i(ad) \rightarrow \Omega^{d-3+i} $$</div>

So for this case, for example, it would take a two-form to a d-minus-three-plus-two or a d-minus-one-form. So curvature is an ad-valued two-form. And, if I had such a shiab operator, it would take ad-valued two-forms to ad-valued d-minus-one-forms, which is exactly the right space to be an \(\alpha\) coming from the derivative of an action.

'''Audience Member:''' Can you just clarify what the index on the shiab is? So you take i-forms to d-minus-three-plus-1?

'''Eric Weinstein:''' Three-plus-i.

What about the torsion? Can we rescue the torsion? Here, again, we have good news. The torsion is problematic, but if I look at a different field—which I'm going to call the '''augmented torsion''', and I define it to be the regular torsion, which would be \(\pi\) minus this expression, this turns out to be beautifully invariant again.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ T_{\epsilon, \pi} = \pi - h^{-1}d_{A_0}h $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon F_\pi +[\bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon T_\omega, T_{\epsilon, \pi}] + *T_{\epsilon, \pi} = 0 $$</div>

[[File:GI-Exact.jpg|center]]

Let's define matter content in the form of Omega-0 tensored in the spinors, which is a fancy way of saying spinors, together with a copy of the one-forms tensored in the spinors. And let me come up with two other copies of the same data—so I'll make \(\Omega^{d-1}\) just by duality, so imagine that there's a Hodge star operator.

So in one case, I can do plus star to pick up the \(A_{\pi}\). But I am also going to have a derivative operator if I just do a star operation, so I need another derivative operator to kill it off here. So I'm going to take minus the derivative with respect to the connection, which defines a connection one-form as well as having the same derivative coming from the Levi-Civita connection on \(U\).

So in other words, I have two derivative operators here. I have two ad-value one-forms. The difference between them has been to be a zeroth-order, and it's going to be precisely the augmented torsion. And that's the same game I'm going to repeat here.

I'm going to do that again on the other side. There are going to be plus and minus signs, but it's a magic bracket that knows whether or not it should be a plus sign or a minus sign and I apologize for that, but I'm not able to keep that straight. And then there's going to be one extra term, where all these \(T\)s have the \(\epsilon\) and \(\pi\)s.

[[File:GU Presentation Sym-Fld-Eq Diagram.png|center]]

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\epsilon d_A \zeta + [\bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\epsilon\zeta, T] + [T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\epsilon\zeta] + *\zeta = F_A \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu + [[T, \nu], T] + [T, [T, \nu]] + [[T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu], T] + *\nu $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ *(d_A^* \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em^* + * + ...)(\bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon F_{A_\pi} + ...) $$</div>

So for example, if \(X\) was four-dimensional, therefore d equals 4, then \(Y\) in this case would be \(d^2\), which would be 16 plus 3d, which would be 12, making 28, divided by two, which would be 14. So in other words, a four-dimensional proto-spacetime—not a spacetime, but a proto-spacetime with no metric—would give rise to a 14-dimensional observerse portion called \(Y\). Now I believe that in the lecture in Oxford I called that \(U\), so I'm sorry for the confusion, but of course, this shifts around every time I take it out of the garage, and that's one of the problems with working on a theory in solitude for many years.

Most fields—and in this case we're going to call the collection of two-tuples \(\omega\), so inside of \(\omega\) that will be, in the first tuple we'll have \(ϵ\) and \(ϖ\), written in sort of a nontraditional variation of how we write this symbol for \(\pi\); in the second tuple, we'll have the letters, \(\nu\) and \(\zeta\), and I would like them not to move because they honor particular people who are important. So most fields, in this case \(\omega\), are dancing on \(Y\), which was called \(U\) in the lecture, unfortunately. But, they are observed via pullback as if they lived on \(X\). In other words, if you're sitting in the stands, you might feel that you're actually literally on the pitch, even though that's not true. So what we've done is we've taken the U of Wheeler, we've put it on its back, and created a double-U structure.

So, there is one way in which we've reversed the fundamental theorem of Riemannian geometry, where a connection on \(X\) leads to a metric on \(Y\). So if we do the full transmission mechanism out, \(ℷ\) on \(X\) leads to \(\aleph_ℷ\) for the Levi-Civita connection on \(X\). \(\aleph_{ℷ}\) leads to \(g_{\aleph}\), which is—sorry, \(g_\aleph\). I'm not used to using Hebrew in math.* So \(g_{\aleph}\), then, is a metric on \(Y\), and that creates a Levi-Civita connection of the metric on the space \(Y\) as well, which then induces one on the spinorial bundles. In sector II, the inhomogeneous gauge group on \(Y\) replaces the Poincaré group and the internal symmetries that are found on \(X\). And in fact, you use a fermionic extension of the inhomogeneous gauge group to replace the supersymmetric Poincaré group, and that would be with the field content zero forms tensored with spinors direct sum one-forms tensored with spinors all up on \(Y\) as the fermionic field content.

So with function spaces, we can take the bundle of groups using the adjoint action of \(H\) on itself and form the associated bundle, and then move to \(C^\infty\) sections to get the so-called gauge group of automorphisms. We have a space of connections, typically denoted by \(A\), and we're going to promote \(\Omega^1(B,\text{ ad}(P_H))\) to a notation of script \(\mathcal{N}\) as the affine group, which acts directly on the space of connections.

And then we have an action of \(G\), that is the inhomogeneous gauge group, on the space of connections, because we have two different ways to act on connections. We can either act by gauge transformations or we can act by affine translations. So putting them together gives us something for the inhomogeneous gauge group to do.

So our summary diagram looks something like this. Take a look at the \(\tau_{A_0}\). We will find a homomorphism of the gauge group into its inhomogeneous extension that isn't simply inclusion under the first factor. We—and I'm realizing that I have the wrong pi projection, that should just be a simple \(\pi\) projecting down. We have a map from the inhomogeneous gauge group via the bi-connection to \(A \times A\), connections cross connections, and that behaves well according to the difference operator \(\delta\) that takes the difference of two connections and gives an honest ad-valued one-form.

We then get to shiab operators. Now, a '''shiab operator''' is a map from the group crossed the ad-valued i-forms. In this case, the particular shiab operator we're interested in is mapping i-forms to d-minus-three-plus-i-forms. So for example, you would map a two-form to d-minus-three-plus-i. So if d, for example, were 14, and i were equal to two, then 14 minus three is equal to 11 plus two is equal to 13. So that would be an ad-valued 14-minus-one-form, which is exactly the right place for something to form a current, that is, the differential of a Lagrangian on the space.

Marcus du Sautoy, Peter Thiel, Isadore Singer, Raoul Bott, Michael Grossberg, Adil Abdulali, Harry and Sophie Rubin, Bret Weinstein and family, Heather Heying, and Zach and Toby, Peter Freyd, Scott Axlerod, Nima Arkani Hamed, Luis Alvarez Gaume, Edward Frankel, Dror Bar Natan, Shlomo Sternberg, David Kazhdan, Daniel Barcay, Karen and Les Weinstein, Haynes Miller, Ralph Gomory, John Tate, Sidney Coleman, Graeme Segal, Robert Hermann, and Hira and Esther Malaney.

I do want to leave you with one thought. I really think that we've gotten completely bent out of shape about trying to formalize and routinize science, and it doesn't work. You cannot mandate science as social engineering, you can't decide that science is always in the zeitgeist and done by committee. In fact, it is essential to understand that science will not conform to what you want. One of the things that I'm very proud of, and I think is quite true, is the saying that great science has the scientific method as its radio edit. I don't think that great science is actually done the way we say it's done, and I think that <a href="https://blogs.scientificamerican.com/guest-blog/the-evolution-of-the-physicists-picture-of-nature/">Dirac's 1963 Scientific American article</a> should be read by absolutely everyone.