Algebras with a nontrivial center, are essential for EEQT. Both physicists and mathematicians, for some reason, have neglected this possibility and approach. Mathematicians do not like these algebras, because they “complicate the matter” – they think that it is an unnecessary complication. For example, in the monograph of AlainConnes on noncommutative geometry we can find a section on “Reduction theory”.
1.β Reduction theory. This theory was developed by von Neumann in 1939, but first published in 1949 [413].
Reduction theory tells us that it is always possible to decompose an algebra into a family of other algebras that have no non-trivial central elements, so called “factors”. Then mathematicians concentrate on “factors”.
But it is much like with atoms: it is true that every molecule can be decomposed into atoms, but a molecule is more than just a collection of atoms, and, as they say, “More Really is Different”. In Mile Gu, Christian Weedbrook, Alvaro Perales, Michael A. Nielsen, “More is different” the authors write:
In 1972, P.W. Anderson suggested that `More is Different', meaning that complex physical systems may exhibit behavior that cannot be understood only in terms of the laws governing their microscopic constituents. We strengthen this claim by proving that many macroscopic observable properties of a simple class of physical systems (the infinite periodic Ising lattice) cannot in general be derived from a microscopic description. This provides evidence that emergent behavior occurs in such systems, and indicates that even if a `theory of everything' governing all microscopic interactions were discovered, the understanding of macroscopic order is likely to require additional insights.
A molecule is a collection of organized atoms. But the organization within a molecule is as important, if not more so, than its atoms that are organized!
A collection of atoms that are not organized represents chaos. The fact that studying the organization of constitutive elements is an important part of physics has been recognized only recently, when physicists and mathematicians started studying nonlinear dynamics and self-organized systems. But this kind of research has not yet reached quantum theory.
We need more of these "factors". In fact we need an infinite number of them, organized by "arrows" between them. But that will become evident later on.
P.S.1. This is for my wife - she likes it. More is different.
Look into my eyes
You will see, what you mean to me
Search your heart, search your soul
When you find me then, you'll search no more
Don't tell me its not worth trying for
You cant tell me its not worth dying for
You know its true, everything I do, I do it for you
Look into your heart, you will find
There is nothing there to hide
Take me as I am, take my life
I would give it all, I would sacrifice
Don't tell me its not worth fighting for
I cant help it, there's nothing I want more
You know its true, everything I do, I do it for you
There is no love, like your love
And no other, could give me more love
There's nowhere, unless you're there
All the time, all the way
You cant tell me its not worth trying for
Just cant help it, there's nothing in the world I want more
I would fight for you, yeah I'd lie for you
Walk the wire for you, yeah I'd die for you
You know its true
Everything I do, I do it for you
P.S.2 The beginning of "Geometry of the conformally compactified Minkowski space"
q = x^2 + y^2 + z^2 - t^2;x1 = x; x2 = y; x3 = z; x4 = t; x5 = (1 - q)/2; x6 = -(1 + q)/2;x46 = Simplify[4*(x6^2 + x4^2)];ulicz = ComplexExpand[Simplify[4*(x6 - I x4) {{x5 + I x3, x2 + I x1}, {-x2 + I x1, x5 - I x3}}]];X = {{t + z, x - I y}, {x + I y, t - z}};cX = Simplify[(X - I IdentityMatrix[2]).Inverse[X + I IdentityMatrix[2]]];Simplify[cX - ulicz/x46]
And we had a little earthquake yesterday:
OK I don't know if this makes sense but is it loosely true that the things not in the center are the bosons and fermions and the things in the center are the differential geometry field? This could make some sense to me from a Clifford algebra/Heisenberg group/Fock space point of view.
ReplyDeleteYes, that is a possibility - roughly.
Delete@John G
DeleteIf we think of fermions and bosons as objects in a certain category, and Clifford algebra, Heisenberg group and Fock space as morphisms (transformations), we can start to think of them in the context of differential geometry as the structure of the space in which these particles exist and act.
This is a very abstract and general approach, but it makes sense on some level.
"I welcome all questions, comments, suggestions concerning these notes. ".
ReplyDeleteThis is an interesting concept. I think densification could be useful as a future description to replace the concept of inflation.
However, I still have one fundamental question that makes it difficult for me to read this text. Quo vadis?
Good question. Thank you. I will soon start to lazily write the Introduction section, sentence after sentence, which, in particular, will aim at partially addressing this question.
Delete"P.S.4. Exercise for the Reader: Find 10 reasons why understanding the conformal symmetry and the conformal compactifications of Minkowski space-time is nowadays the most important thing to do for a theoretical physicist? Write your answers.".
ReplyDelete1) As part of their efforts to reconcile quantum mechanics with general relativity, many physicists are researching quantum gravity. Understanding conformal symmetry is essential in this context, as it potentially allows for geometric interpretations of quantum field theories.
2) Conformal symmetry and compactification can provide new insights into the nature of time. From the perspective of Minkowski space-time, time is seen as a fourth dimension, which can provide a more unified understanding of the Universe.
3) Conformal symmetry plays a significant role in quantum field theory. It allows for a better understanding of the properties of quantum fields, which is central to the modern standard model of particle physics.
4) Understanding the conformal compactifications of Minkowski space-time can provide a solid framework for building category theoretic models in physics, which can describe complex phenomena in a universal language.
5) Unifying Physics: Theoretical physicists constantly seek a "Theory of Everything" that will unify all known physical phenomena. The understanding of conformal symmetry could provide the necessary insights and tools to help develop this theory.
6) Conformal compactifications of Minkowski space-time can help to explain the large-scale structure of the Universe, and even the Big Bang. Such an understanding is crucial in exploring the history and future of our universe.
7) Conformal symmetry, when integrated with other symmetries, can be used to predict the behavior of particles under different conditions. This could lead to the discovery of new physical phenomena or particles.
8) Understanding how symmetry breaking occurs in a conformally symmetric system could help in understanding other symmetry-breaking phenomena in physics, such as those that are believed to occur during phase transitions in the early Universe (I see here an interesting application to EEQT).
9) In recent years, there's been interest in the application of geometric and topological methods in neurobiology. Conformal symmetry can provide insights into the complex neural networks that define our cognitive abilities. For instance, the compactification of Minkowski space-time could be used as a model to understand the propagation of signals within the brain, and how time perception is maintained across different scales.
10) The symmetries inherent in physical laws can also be found in music and are key to our sense of harmony and balance. The understanding of such symmetries might therefore lead to innovative new structures and forms in music composition.
Each of these points can, of course, be developed further. Here I have only described it briefly.
1. The conformal group is the full symmetry group of Maxwell's equations.
ReplyDelete2. The conformal group is the full symmetry group of the Dirac equation.
3. Bradonjic's model for general relativity has a conformal metric which I think of as having the conformal group as its g-structure group.
4. The Lorentz group and deSitter group have been well looked at without a lot of progress lately and the conformal group would be the next logical one to look at.
5. Even mainstream models like string theory have no problem with extra dimensions and just two extra dimensions get you to the conformal group.
6. Compactification with this conformal group for the well accepted Minkowski spacetime could tame infinities as well as use the conformal group.
7. Conformal symmetry is important for light cone modeling.
8. Conformal symmetry relates to twistors which could be useful structures.
9. The conformal group root vector polyhedron perhaps shows analogies between physics and personality circumplex models (my original reason for being interested in your early 90s conformal group work).
10. Conformal structures perhaps show an analogy between cellular automata symmetry and symmetry in physics (my original reason for getting interested in differential geometry).
So combining these last two comments we have already 20 different reasons!
ReplyDeleteAdding now the conformal group to the pdf notes.
"P.S.11. Exercise for the Reader: prove both statements in Remark 1 in the pdf file.".
ReplyDeleteAd 1) Proving that the transformation X -> -X is in SO(4, 2)
Let's assume that X is any element of R^6, and -X is the element obtained by changing the sign of each element in X. Then we can define L as the diagonal matrix with -1 on the diagonal, which multiplied by the vector X gives us -X. This means that L is the transformation matrix that transforms X into -X. We need to show that L satisfies the condition L^{T}GL = G.
We have:
L^T = transpose of L = L (since L is a diagonal matrix)
So L^{T}GL = LG = L (since G is an identity matrix with -1 in some diagonal positions)
So, L is an element of O(4,2). As L has determinant -1 to the power of 6 (because it is six-dimensional), which gives 1, it is also an element of SO(4,2).
Ad 2) Proving that the transformation acts on \tilde{M} as an identity map.
ReplyDeleteWe assume that \tilde{M} is the projection space of N, i.e. [X] belongs to \tilde{M} if X is different than 0 and Q(X) = 0. Where Q(X) is the quadratic form of X, Q(X) = G^T XG. Then for [X] belonging to \tilde{M}, we have [X] = [-X], because both X and -X are elements of N, which are equivalent. Therefore L[X] = [-X] = [X], showing that the action of L on \tilde{M} is the identity map.
Both reasonings of Mathilde S. were detailed (even 'over-detailed') and correct.
ReplyDeleteThe Exercise in the file is still waiting for someone to do it.
"P.S.12. 13:04 Added beginning of Section 5 and Exercise 1 at the end.".
ReplyDeleteThe topology principles involved in this exercise might be a bit complex, but I'll try to simplify them as much as I can.
Ad 1) Prove that $\hat{M}$ is connected and path-connected:
Topologically, a set is considered connected if we can't divide it into two disjoint, non-empty open subsets. $\hat{M}$ is the compactified Minkowski space, which is derived from the Minkowski space (which is connected) by a compactification process that adds 'points at infinity' without splitting the existing points. Therefore, we can say that $\hat{M}$ is connected.
A space is path-connected if there exists a continuous path between any two points in the space. The same logic here - the original Minkowski space is path-connected, and the compactification process doesn't disrupt this path connectivity but extends it to the 'points at infinity'. Hence, $\hat{M}$ is also path-connected.
Ad 2) Calculate the homotopy group $\pi_1(\hat{M})$:
The first homotopy group, $\pi_1$, of a topological space refers to its fundamental group, which, in simple terms, represents different 'loops' in the space that can't be deformed into each other through a continuous transformation.
In this case, $\hat{M}$ being a compactified Minkowski space is topologically equivalent to the 4-dimensional sphere $S^4$ (since compactification is essentially 'one-point compactification' adding a single point at infinity). The fundamental group of an n-sphere is known to be trivial (i.e. consisting only of the identity element) for n greater than or equal to 2. Hence, the homotopy group $\pi_1(\hat{M})$ is trivial.
Both reasonings above have errors.
ReplyDeleteNew Exercise: find all these errors and correct them.
Is this corrected reasoing acceptable?
DeleteAd 1) We defined $\hat{M}$ as the product of spheres $S^3 \times S^1$. Both $S^3$ and $S^1$ are connected and path-connected spaces. The product of connected spaces is connected, and the product of path-connected spaces is path-connected. Therefore, $\hat{M}$, being $S^3 \times S^1$, is both connected and path-connected.
I will reflect on Ad 2) in a few moments. If you have comments on Ad 1) then ask me specific questions. Please.
I will wait for these comments before returning to Ad 2), as the essence of the error may be the same.
The above is correct.
ReplyDeleteAd 2) corrected reasoning:
DeleteAs per the definition of the homotopy group, $\pi_1(X)$ is the set of path-components (loops up to deformation) in $X$ with a chosen base point.
The fundamental group of a product of spaces is the product of the fundamental groups of the spaces, i.e., $\pi_1(X \times Y) = \pi_1(X) \times \pi_1(Y)$. We know that $\pi_1(S^3)$ is trivial since $S^3$ is simply connected, and $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$ (the group of integers under addition), because any loop in $S^1$ can be wound around the circle an integer number of times (in this issue I am not convinient and I know it).
Therefore, the fundamental group $\pi_1(\hat{M})$ would be ${e} \times \mathbb{Z} \cong \mathbb{Z}$, where $\cong$ denotes a group isomorphism, and $e$ is the identity element.
It is possible that I cannot see the specifics of this. I am looking at it quite generally and something may be missing.
I'm not sure I'm spelling it right. The second point is harder for me than the first and I don't feel quite right, I'm relying on intuitions which are relative and uncertain at the moment. I don't have an epiphany here.
Now also the second point is done correctly.
ReplyDeleteBjab -> Ark
ReplyDelete"Then PN , with its projective topology, is a compact projective quadric."
Co to jest quadric?
Search Google for "projective quadric". I searched and found this:
DeleteDefinition 9 A quadric in a projective space P(V ) is the set of points whose rep- resentative vectors satisfy B(v, v)=0 where B is a symmetric bilinear form on V . The quadric is said to be nonsingular if B is nondegenerate. The dimension of the quadric is dimP(V ) − 1.
So, our quadric is "nonsingular" - good to know.
Bajb ->
DeleteArk
Iluwymiarowy jest ten PN quadric?
N is 5-dimensional in a 6-dimesional space. PN is one dimension less - thus 4-dimensional.
ReplyDeleteBjab -> Ark
DeleteDziękję.
Bjab -> Ark:
ReplyDelete""Let c be one of the two square roots of det(U), so that c
2 =
1/ det(U)."
Nie rozumiem tego. (Może zabrakło sprzężenia?)
Indeed. Corrected. Thank you!
DeleteBjab -> Ark:
Delete"Corrected"
I can't see.
You will need to reload the page. Changed to
DeleteLet c be one of the two square roots of 1/ det(U)