LOG#240. (Super)Dimensions.


Hi, everyone! Sorry for the delay! I have returned. Even in this weird pandemic world…I have to survive. Before the blog post today, some news:

  1. Changes are coming in this blog. Whenever I post the special 250th post, the format and maybe the framework will change. I am planning to post directly in .pdf format, much like a true research paper.
  2. I survive, even if you don’t know it, as High School teacher. Not my higher dream, but it pays the bills. If you want to help me, consider a donation.
  3. Beyond the donation, I am aiming to offer some extra stuff in this blog: free notes (and links) to my students or readers, PLUS, a customized version of them that of course you should pay me for the effort to do. It would me help me to sustain the posting or even managing independence of my other job that carries from me time to post more often.
  4. I will offer a service of science consulting to writers, movie makers, and other artists who wish for a more detailed scientific oriented guide.
  5. I will send the full bunch of my 250 blog posts as soon as possible, with customized versions if paid. 1 euro/dollar per blog post will be my price. The customized election of blog posts will be negotiated later, but maybe I will offer 25 blog posts packs as well, plus the edition cost. Expensive? Well, note that I had to put lot of effort to build this site alone. I need to increase income in these crisis times. You will be able to find the posts here for free anyway, but if you want them edited, you can help me further. If I could I would leave my current job since I am unhappy with it and with COVID19 is a risk to be a teacher (if presence is required into class of course).

Topic today are dimensions. Dimension is a curious concept. Fractal geometry has changed what we used to consider about dimensions, since fractals can have non-integer dimensions. Even, from certain viewpoint, you can also consider negative dimensions, complex dimensions and higher versions of it. With fractals, you have several generalized dimensions:

(1)   \begin{equation*} \tcboxmath{D_{box}=D_0=-\lim_{\varepsilon\rightarrow 0}\left(\dfrac{\log N(\varepsilon)}{\log\dfrac{1}{\varepsilon}}\right)} \end{equation*}

Next, information dimension

(2)   \begin{equation*} \tcboxmath{ D_1=\lim_{\varepsilon\rightarrow 0}\left[-\dfrac{\log p_\varepsilon}{\log\dfrac{1}{\varepsilon}}\right]} \end{equation*}

Generalized Renyi dimensions are next

(3)   \begin{equation*} \tcboxmath{D_\alpha=\lim_{\varepsilon\rightarrow 0}\dfrac{\dfrac{1}{\alpha-1}\log \sum p_i^\alpha}{\log\varepsilon}} \end{equation*}

Now, we can also define the Higuchi dimension:

(4)   \begin{equation*} \tcboxmath{ D_h=\dfrac{d \log (L(X))}{d\log (k)}} \end{equation*}

Of course, you also have the celebrated Hausdorff dimension

(5)   \begin{equation*} \tcboxmath{\mbox{dim}_h(X)=\mbox{inf}\left{d\geq 0: C_H^d(X)=0\right}} \end{equation*}

In manifold theories, you can also define the codimension:

(6)   \begin{equation*} \tcboxmath{\mbox{codim}(W)=\mbox{dim}(V)-\mbox{dim}(W)=\mbox{dim}\left(\dfrac{V}{W}\right)} \end{equation*}

if W is a submanifold W\subseteq V. Also, if N is a submanifold in M, you also have

(7)   \begin{equation*} \mbox{codim}(N)=\mbox{dim}(M)-\mbox{dim}(N) \end{equation*}

such as

(8)   \begin{equation*} \tcboxmath{\mbox{codim}(W)=\mbox{codim}\left(\dfrac{V}{W}\right)=\mbox{dim}\left(\mbox{coker}(W\rightarrow V)\right)\right)} \end{equation*}

Finally, superdimensions! In superspace (I will not go into superhyperspaces today!), you have local coordinates

(9)   \begin{equation*} \tcboxmath{X=(x,\Xi)=(x^\mu, \Xi^\alpha)=(x^\mu, \theta,\overline{\theta})} \end{equation*}

where \mu=0,1,2,\ldots, n-1 and \alpha=1,2,\ldots,\nu. Generally, \nu=2m, so the superdimension is the pair (n,\nu)=(n,2m) in general. In C-spaces (Clifford spaces) you have the expansion in local coordinates:

(10)   \begin{equation*} \tcboxmath{X=X^A\gamma_A=\left(\tau, X^\mu,X^{\mu_1\mu_2},\ldots,X^{\mu_1\dots\mu_D}\right)} \end{equation*}

and if you go into C-superspaces, you will also get

(11)   \begin{equation*} \tcboxmath{Z=Z^W\Gamma_W=(X^A; \Xi^\Omega)=\left(\tau,X^\mu,X^{\mu_1\mu_2},\ldots,X^{\mu_1\dots\mu_D}; \theta, \theta^\alpha,\theta^{\alpha_1\alpha_2},\ldots,\theta^{\alpha_1\ldots\alpha_m}\right)} \end{equation*}

With superdimensions, you can also have superdimensional gauge fields and supermetric fields, at least in principle (in practice, it is hard to build up interacting field theories with higher spins at current time). For supergauge fields, you get

(12)   \begin{equation*} \tcboxmath{A=A^W\Gamma_W=(A^Z; \Xi^\Omega)=\left(\tau,A^\mu,A^{\mu_1\mu_2},\ldots,A^{\mu_1\dots\mu_D}; \Theta, \Theta^\alpha,\Theta^{\alpha_1\alpha_2},\ldots,\Theta^{\alpha_1\ldots\alpha_m}\right)} \end{equation*}

The C-space metric reads

(13)   \begin{equation*} \tcboxmath{ds^2=dX_AdX^A=d\tau^2+dx^\mu dx_\mu+dx^{\mu_1\mu_2}dx_{\mu_1\mu_2}+\cdots dx^{\mu_1\cdots \mu_D}dx_{\mu_1\cdots\mu_D}} \end{equation*}

and more elaborated formula for C-supermetrics and C-superhypermetric could be done (I am not done with them yet…). The mixed type of gauge fields in C-superspaces (even C-superhyperspaces) is yet hard to even myself. Work for another day!

Definition 1 (UR or eTHOR conjecture).

There is an unknown extended theory of relativity (eTHOR), ultimate relativity (UR), and it provides transformation rules between any type of field (scalar, spinorial,vector, tensor, vector spinor, tensor spinor, and general multitensor/multiform multispinor) and their full set of symmetries. Consequences of the conjecture:

  • UR involves coherent theories of higher spins AND higher derivatives, such as there is a full set of limits/bound on the values of the n-th derivatives, even those being negatives (integrals!).

  • UR involves a generalized and extended version of relativity, quantum theory and the equivalence principle.

  • UR provides the limits of the ultimate knowledge in the (Multi)(Uni)verse, even beyond the Planck scale.

  • UR will clarify the origin of space-time, fields, quantum mechanics, QFT and the wave function collapse.

  • UR will produce an explanation of M-theory and superstring theory, the theory of (D)-p-branes and the final fate of the space-time singularities, black hole information and black hole evaporation, and the whole Universe.

See you in other blog post!

P.S.: Please, if you want to help me, I wish you can either donate or buy my stuff in the near future. My shop will be launching soon,…In September I wish…

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