First we will review of definitions and results from previous posts.
We
have discussed indefinite metric complex vector space X, endowed
with a scalar product (z,z') of signature (n,m), where m,n ≥ 1. Let
ei (i=1,2,...,n) be a basis in X. Then each z in X can be decomposed
into the basis vectors
z
= z1
e1+...+zm+n
em+n
We
call the basis orthonormal if the scalar product, when written in
this basis, takes the form
(z,z')
= - z1*z'1
-...-
zm*z'm
-.+
zm+1*z'm+1
+...+
zn*z'n
, (*)
where
zi*
denotes the complex conjugate to zi.
Selecting
an orthonormal basis in X, we identify X with Cm+n,
= Cm⊕Cn.
The vectors in X we then write as columns z={w,v}, with w ∈ Cm
and
v ∈ Cn.
The scalar product (z,z') takes then the form
(z,z')
= -w*w' + v*v'
where
* applied to vectors in Cm
and
Cn
denotes the hermitian conjugate.
We
denote by J the set of all maximal positive subspaces of X. A
subspace V of X is called positive if the scalar product (z,z')
restricted to V is positive definite. By Sylvester's Law of Inertia
each such V is necessarily n-dimensional. We are interested in the
set J of all such subspaces. In a convenient parametrization by mxn complex matrices Z with Z*Z<I (that we will introduce below), the pseudo-unitary group U(n,m) will act on J by (generalized) linear fractional transformations. In mathematics J is an example of a bounded symmetric domain of type I(one).
Note: There are also infinite-dimensional generalizations - see "Bounded Symmetric Domains in Banach spaces" by Cho-Ho Chu. We will restrict ourselves to the finite-dimensional case.
Strictly speaking we are interested in the group U of all isometries of X endowed with the scalar product (z,z'). But once we have selected an orthonormal basis of X, then U becomes identified with U(n,m) - the group of all complex (m+n)x(m+n) matrices preserving the scalar product (*). Introducing the diagonal block matrix J0
J0= diag(-Im, In) ,
we have written the condition on matrices U from U(n,m) as
U†J0 U = J0
where the dagger † denotes the hermitian conjugate. We notice that J0 itself is in U(n,m).
Note: We are using bold letters to denote (m+n)x(m+n) matrices.
We have defined J is as the set of all maximal positive subspaces of X. Equivalently we could have defined J as the set of all linear operators J acting on X satisfying the three conditions:
1) J=J*,
2) J2=I,
3) the sesquilinear form (z,z')J defined by
(z,z')J =(z,Jz')
is positive definite.
Note: Notice that J0 ∈ J.
If V is a maximal positive subspace of X, and if W is its orthogonal complement, then J corresponding to V is defined as the unique linear operator defined as the identity on V and as minus identity on V. Conversely, if J satisfies the conditions 1),2),3), then its eigenspace belonging to the eigenvalue +1 is a maximal positive subspace of V. This follows by an elementary linear algebra.
We have shown that every J ∈ J is of the form :
Where Z is an mxn matrix satisfying Z*Z<I (which is equivalent to ZZ*<I) uniquely determined by J. Moreover the maximal positive subspace determined by J (that is the eigensubspace belonging to the eigenvalue +1) consists of all vectors z of the form
Since this should hold now for any v, comparing with the prvious expression for z' we get
And this is our final formula - a generalized linear fractional transformation. It automatically follows that if Z*Z<I, then Z'*Z'<I.
In the next post we will discuss what happens to this formula when we leave the safe ground and try to do something "forbidden", namely extend the above transformation formula to Z such that Z*Z=I. For m=n=2 such Z parametrize points of the "Shilov boundary" - the compactified Minkowski spacetime of events equipped with the flat conformal causal (light-cone) structure.
P.S.1. Everything presented in this note requires only elementary linear algebra. In particular I did not use any computer algebra software, like for instance Mathematica, or Reduce, which I love to use when it helps.
Well, I used one line of code (which I am not particularly proud about) to get the formula (8) from The Sound of Silence:
Reduce[2 x + x y + 2 x Sqrt[1 + y] == y, y]
The function graph above this formula comes with the code.
P.S.2 (31-12-22) Started reading Christopher Langan's "Introduction to Quantum Metamechanics". Observations from the first page: Langan rightly complains about the state of quantum mechanics. Mentions the need for "post-quantum mechanics" (post-QM). (I think he borrowed this term from Jack Sarfatti?) But then Langan writes about it: "Because this theory is necessarily a metatheory (or theoretical metalanguage) of QM, it is called Quantum Metamechanics or QMM)."
I do not see any necessity for post-QM to be a metatheory. What I see is the necessity of having a better theory than the standard QM.
P.S.3 31-12-22 13:00 Encouraged by Irina Eganova I have started reading "World as Space and Time" by <a href="https://en.wikipedia.org/wiki/Friedmann_equations">A.A. Friedman</a>. Beautifully written! The book (in Russian) is accessible for reading online <a href="https://reallib.org/reader?file=583994&pg=7">here</a>
P.S.4 Searching the net for Friedman and "space-time boundary" I have stumbled upon "Category Theory in Physics, Mathematics and Philosphy", Ed. Marek Kuś and Bartłomiej Skowron, Springer 2019, and there the paper by Michael Heller and Jerzy Król "Beyond the Space-Time Boundary". Interesting reading though only superficially related to my own projects. The authors claim that " The standard geometric tools on M do not allow one “to cross the boundary”. Well it all depends on what they call "standard".
Happy new Year! May your dreams come true! But while chasing your dreams pay close attention to reality left and right!
Could you write in the notes a little more about how these formulas can be applied? I'm interested in how you see it, what you would actually like to describe and what your associations are with these formulas.
ReplyDeleteSome of such applications can be found on web pages of Tony Smith, for instance here:
DeleteBounded Complex Domains
Some other are mentioned in the paper by K. Korotkov, A. Levichev: "The 3-fold Way and Consciousness Studies" that you can download from the Chronos web page Природа времени
In my comment to the previous post I wrote about a paper by I.A. Eganova and W. Kallies describing experimental evidence for "action at a distance". I have asked the first author about some hints concerning a theoretical framework that She has in mind when thinking about these effects. This morning in reply to my question She kindly attached the paper stored on arxiv:
ReplyDeleteI.A. Eganova and W. Kallies, "A Special Physical Phenomenon — Innate
Interconnection of Space-time Points"
Here is the Abstract:
In the light of A.A. Friedman's conceptual analysis of the World of events as a mathematical model of the physical reality in his book "The World as Space and Time", a priory, innate interconnection of events belonging to one and the same moment of time, which can condition the space-time metric, is considered with a summary review of its astronomical observations by N.A.Kozyrev's method.
There is also twice as long version of this paper in Russian:
DeleteОСНОВАНИЕ МИРА МИНКОВСКОГО
КАК МАТЕМАТИЧЕСКОЙ СТРУКТУРЫ: К ОТВЕТУ
НА ВОПРОС РИМАНА
Thank you very much. Now I just need to come up with something regarding gravitational effects on the Planck scale. I am scheduled to speak about this at a conference for physicists in February. I see a problem that I need to address on the ground of category theory.
DeleteA conceptual difficulty in combining quantum mechanics with general relativity arises from the contrasting role of time within these two frameworks. In quantum theories time acts as an independent background through which states evolve, with the Hamiltonian operator acting as the generator of infinitesimal translations of quantum states through time. In contrast, general relativity treats time as a dynamical variable which relates directly with matter and moreover requires the Hamiltonian constraint to vanish.
I already know what I'm going to talk about. About the problem of time. Nevertheless, I am personally appalled by what I read on Wikipedia. Take a look at this article: https://en.wikipedia.org/wiki/Problem_of_time
And there:
„For macroscopic systems the directionality of time is directly linked to first principles such as the second law of thermodynamics.”.
This is an Eddington's concept that the author himself never explained well enough. To put it briefly, it was stated that there is a correlation between time and entropy as both quantities increase.
Well, there are many correlations in this world. It is important to keep in mind that the existence of correlations can be coincidental. In the case of time and entropy, Eddington naively assumed that both quantities were increasing. What about the entropy of black holes? What about the event horizon? What about quantum entanglement?
And modern physicists read something like this and just duplicate it.
Assuming the truth of the hypothesis of inflation of the Universe (which most physicists assume) additionally, it may be worth noting that if the Universe is expanding then the maximum value of entropy is also increasing. And besides, why is it usually said only about the expansion of space? Why do we not talk about the expansion of time?
I believe Einstein had the idea, but did not complete work on it. Teleparallelism was an attempt to capture certain effects, such a theory would have a wider range of applicability than general relativity, but it would probably not take into account gravitational effects on small scales. At the moment, it seems to me that the only valid approach in theories like quantum gravity is category theory. This is because there is already confusion at the category level.
Confusion of categories and confusion of everything. As a result, people try to combine several inconsistent descriptions and sometimes something will come out of it. This is how the development of so-called quantum gravity is today. Physicists are developing a theory that is wrong at its core.
From the mail: Academia.edu informed be about the 2002 paper H. Cassini, Finite dimensions and the covariant entropy bound. The author is quoting there my old paper with W. Cegla on the causal logic of the Minkowski space of spacetime events. Which is rather closely related to the paper by Irina Eganova quoted in my previous comment.
ReplyDeleteThank you. I was also very interested in this article: https://www.mdpi.com/1099-4300/22/2/247
ReplyDeleteI am analyzing it in more detail today.
Paola Zizzi, who is mentioned in that article Ark posted earlier, seems quite good with beginning entropy, the cosmological constant, and hyperinflation and thus could be good for gravity as the energy first drops below the Planck energy.
DeleteEntropy can be fundamental at the beginning or for a closed system before decoherence but beyond that like you said, it doesn't seem good for time in the fundamental sense since it's just a statistical thing.
For a photon in the Einstein sense, it's whole worldline from its point of view would happen all at once and time-like is kind of just a phase, the propagator phase, which works in the Feynman sense too.
Course that mass thing Ark is talking about hints at other complications like things already connected (conformally) on the brane and thus there's perhaps propagator phase changes on the brane vs propagator phase change with change of brane states.
For actually at the Planck scale/energy before hyperinflation, it may have the overall unbroken symmetry of physics but working with it might be more like figuring out how thoughts are encoded in the brain than physics in the usual sense.
"P.S.1. Everything presented in this note requires only elementary linear algebra. In particular I did not use any computer algebra software, like for instance Mathematica, or Reduce, which I love to use when it helps.".
ReplyDeleteThank you. I was wondering if it is possible to do e.g. block matrices etc. in Mathematica. Especially since I saw that linear fractional transformation is used in hyperbolic geometry, among other things. I also read this: https://mathematica.stackexchange.com/questions/8654/creating-diagrams-for-category-theory
Another paper sent to me by academi.edu. Also discussing entropy and in particular black hole entropy, where my old papewr with W. Cegla is being quoted:
ReplyDeleteGeometric entropy, area, and strong subadditivity
published in Classical and Quantum Gravity, Volume 21, Number 9 DOI 10.1088/0264-9381/21/9/011
"I do not see any necessity for post-QM to be a metatheory. What I see is the necessity of having a better theory than the standard QM.".
ReplyDeleteI also don't see the need for QM to be a metatheory. However, if we are talking about a theory that is supposed to unify quantum theory and general relativity theory, here there is rather a clear need for the construction of a metatheory.
However, this raises the question of what a metatheory actually is. Physics is not internally consistent, and its biggest problem is probably time.
The problem of time is a very significant conceptual conflict between general relativity and quantum mechanics. While quantum mechanics regards the flow of time as universal and absolute, general relativity regards the flow of time as malleable and relative.
What exactly is this time according to physics? A background of events, a parameter, an operator? And what actually evolves? Observables or perhaps states? Does it depend on the accepted picture (Schrodinger's or Heisenberg's) and that is to be satisfied?
There is not even agreement on the category under which we describe time. Is that category Ens, Metr, Top, Gr? Maybe something else? And what is time in this type of description? An object, a morphism, one or the other?
Actually, it is not known... These approaches are many and cannot be simply reconciled. Descriptions of time cause paradoxes, and the most relevant conceptual gaps are also related to time. But how below the metatheoretical level are we to create relativistic-quantum descriptions?
Currently, I see hope in category theory and functors. Some approaches to quantum gravity draw on category theory, but I think these attempts are relatively shallow. Descriptions are still inconsistent. However, I think it is worth looking at these descriptions of specific phenomena or particles, then the inconsistencies become clearer and clearer.
One can also look at it somewhat theologically. E.g., corpuscular-wave dualism can be seen as manifestations of a single meta-phenomenon (it's a bit like the duality of Jesus' nature in theology). What kind of phenomenon is it? I don't know that yet, but I think it's worth approaching it from the category theory side and focusing on time, consciousness, etc. in the process.
In doing so, it can be approached using the method from the detail to the whole. The detail here could be, for example, mathematical analysis of individual gravitational effects on the Planck scale (this is a typical issue for some minor physics paper). Applying a certain method in simpler cases can help build intuition.
@Ark
ReplyDeleteCategory theory is therefore so amazing because it looks at mathematics the way higher consciousness looks at the temporal world. At the same time, this higher consciousness does not determine the mechanics of the temporal world.
The mechanics of the temporal world is homomorphic with logic.
At the level of the substance structure of the world, we are dealing with functional analysis, with algebra, with statistics. Slightly deeper is set theory.
At the meta-level, however, we take away the belonging relation and replace it with a morphism. We take away the set. We replace it with a more abstract object.
The most beautiful and intimidating is the quotient category. In the case of groups, it is very easy to operate on it. Other cases are rather difficult.
There is also an example in which the equivalence classes of morphisms are homotopy classes of continuous maps.
I wonder about something. Would you be able to write a note in which you try to reformulate some of your calculations into category theory terms? If you want, it doesn't have to be your own results. I am interested in how you see physics from the level of category theory. Something like this.
@Ark
ReplyDelete"The authors claim that " The standard geometric tools on M do not allow one “to cross the boundary”. Well it all depends on what they call "standard".".
Expand on this thought, please.
I did some google searches to see if there were any category theory papers related to the context in which I see Ark's conformal group work over the years and I found one though not surprisingly I understand none of the category theory.
ReplyDeletehttps://arxiv.org/pdf/1309.6049.pdf
Here's the introduction:
This paper is a sequel to [18] in which the author categorified a finite-dimensional Clifford algebra. The goal of this paper is to categorify an infinite-dimensional Clifford algebra ClZ via a diagrammatic monoidal DG category CL and give a categorical action of CL on a category DGP(R) which lifts the Fock space representation V of ClZ. (See Section 1.1 for precise definitions of ClZ and V .) The main improvement over [18] is that the categorification is done diagrammatically. In particular, the category CL is strict monoidal.
The Clifford algebra Cl and its cousin, the Heisenberg algebra H, arise from the study of commutation relations in fermionic and bosonic quantum mechanics, respectively. Here H has generators
pi, qi, for i in some infinite set I and commutator relations [pi, qj ] = piqj − qjpi = δi,j1, [pi, pj ] = 0, [qi, qj ] = 0. The symmetric algebra of infinitely many variables indexed by I admits a natural action of H.
Khovanov [15] provided a graphical calculus for a categorification of H. More generally, Cautis and Licata [5] presented a categorification HΓ of the Heisenberg algebra HΓ associated to a finite subgroup Γ ⊂ SL2(C). In a later paper [6], they provided 2-representations of quantum affine algebras via complexes in the 2-representation of quantum Heisenberg algebra. The diagrammatic approach to categorification was pioneered by Lauda in [16] where he obtained a categorification of the quantum group Uq(sl2) .
The Clifford algebra Cl is a C-algebra with generators ψj , ψ∗j
for j ∈ Z and relations: {ψi, ψ∗j} = ψiψ∗j + ψ∗jψi = δij , {ψi, ψj} = 0, {ψ
∗i, ψ∗j} = 0, for i, j ∈ Z. It is the analogue of H obtained by replacing the commutation relations by the anticommutation relations. The Clifford algebra Cl acts on the exterior algebra of infinitely many
variables, which is called the Fock space of free fermions. Note that this Fock space also admits an action of the Heisenberg algebra H under the boson-fermion correspondence [11, Section 14.9].
The antisymmetric property of fermions in physics can be understood in the framework of differential graded categories in mathematics. Lipshitz, Ozsv´ath and Thurston in [17] defined a diagrammatic differential graded algebra, called the strands algebra in the context of bordered Heegaard Floer homology. Motivated by the strands algebra, Khovanov [14] gave a diagrammatic categorification of the positive part of Uq(gl(1|2)). A connection between Heegaard Floer homology and 3-dimensional contact topology on the categorical level was observed by Zarev in [20]. Based on this connection the author categorified Ut(sl(1|1)) and its tensor product representations in [19]. Our graphical calculus in this paper is another attempt to pursue these diagrammatic methods for the Clifford algebra.
There was a period of time when I was playing with similar objects. For instance my paper with Coquereaux and Kastler Differential and Integral Geometry of Grassmann Algegras, Reviews in Mathematical Physics 3(01):63-99 DOI: 10.1142/S0129055X91000035 with the following abstract:
DeleteWe give a self-contained exposition of the differential geometry of Grassmann algebras. We also study elementary properties of these algebras from the point of view of Hochschild and cyclic cohomologies.
@Ark
DeleteWhat is actually the physical interpretation of Grassmann algebras? On page 79. you write about creation and annihilation operators. I understand it like this:
a^\dagger -> \theta
a -> d/d\theta
Then (2.13a) can be understood as
{a_i,a_j} = 0, where {,} is an anticommutator,
(2.20) as
{a_i^\dagger, a_j^\dagger} = 0,
(2.21) as
a_i^\dagger a_j + a_j a_i^\dagger = delta^i_j.
The particle number operator could also be defined in terms of Grassmann algebras. So I understand that they involve statistical aspects for fermions.
Can it be understood this way?
@Ark
ReplyDeleteAt the end of this paper you have \epsilon. Can we write that in general:
\epsilon_{i_1},…{i_n} = d/d\theta_n … d/d\theta_1 \theta_{i_1}…\theta_{i_n} ?
In fact, from this you can move very interestingly to braided geometry. But that's not all. I read in passing about twistor spaces. They write about CP3, for example, but I was wondering about octonions. Has anyone tried to describe twistors using M4xCP2? I see it as a twisting space-time surface. The question is whether there are such approaches to twistors.
ReplyDelete