Minkowski Space

The time evolution of unstable SO(3) monopole and dyon classical field configurations is studied numerically. It is found
that as time goes on, a vacuum region of a finite radius r = t develops, and the monopole core becomes a spherical shell of
almost fixed thickness moving out at the speed of light. The monopole and dyon far fields persist outside the shell, with no
indication of the emission of the nonabelian radiation expected as a manifestation of the instability under small perturbations
in the far field region. Even an initially-growing perturbation eventually stabilizes and is split into a part that moves out to
spatial infinity and another that moves in, is reflected in the vicinity of the monopole center, and then moves out. The
general behavior is very similar to that of certain Minkowski-space, spherically-symmetric classical solutions discovered a
few years ago. The stability of those solutions is also studied.

People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the Ancestors knew all seven. // Люди ныне не знают Вселенной как Целого, не зная грани меж Этим и Тем. Тайна  ее есть тайна забытого тела, седьмого в строю многогранных как Жизнь, Семь. Мы знаем шесть ныне. А Пращуры знали все семь.

The ability of Minkowski Functionals to characterize local structure in different biological tissue types has been demonstrated in a variety of medical image processing tasks. We introduce anisotropic Minkowski Functionals (AMFs) as a novel variant that captures the inherent anisotropy of the underlying gray-level structures. To quantify the anisotropy characterized by our approach, we further introduce a method to compute a quantitative measure motivated by a technique utilized in MR diffusion tensor imaging, namely fractional anisotropy. We showcase the applicability of our method in the research context of characterizing the local structure properties of trabecular bone micro-architecture in the proximal femur as visualized on multi-detector CT. To this end, AMFs were computed locally for each pixel of ROIs extracted from the head, neck and trochanter regions. Fractional anisotropy was then used to quantify the local anisotropy of the trabecular structures found in these ROIs and to compare its distribution in different anatomical regions. Our results suggest a significantly greater concentration of anisotropic trabecular structures in the head and neck regions when compared to the trochanter region (p < 10-4). We also evaluated the ability of such AMFs to predict bone strength in the femoral head of proximal femur specimens obtained from 50 donors. Our results suggest that such AMFs, when used in conjunction with multi-regression models, can outperform more conventional features such as BMD in predicting failure load. We conclude that such anisotropic Minkowski Functionals can capture valuable information regarding directional attributes of local structure, which may be useful in a wide scope of biomedical imaging applications.

The goal of the present paper is to reconstruct the history of Minkowski spacetime, focusing on the specific understanding that Minkowski had of his own model in the context of his worldview. To achieve it, we will first take a look at Minkowski’s personal academic history. By exploring his scientific development, certain key elements in his approach to mathematics and physics will be highlighted, so that his worldview is put into perspective and supported by evidence. Afterwards, a brief summary of Einstein’s work on special relativity is presented, with the aim to speculate on Einstein’s own way of interpreting spacetime. This will also set up the foundation for a comparison with Minkowski. Then, we will analyze the main lectures through which Minkowski presented his geometric model of spacetime to the German scientific community, focusing especially on Raum und Zeit. Here, Minkowski’s worldview is further articulated. Last but not least, we will endorse that, despite Minkowski’s work becoming crucial for Einstein and Einstein's interpretation of spacetime most likely growing closer to Minkowski's, the two scientists ultimately shared crucial differences in weltanschauung.

We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwinger's Master Theorem is commented in connection with closure relations. An adaptation of the Born's Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the 8-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.

This paper presents a brief review of the newly developed \emph{Extended Electrodynamics}. The relativistic and non-relativistic approaches to the extension of Maxwell equations are considered briefly, and the further study is carried out in relativistic terms in Minkowski space-time. The non-linear vacuum solutions are considered and fully described. It is specially pointed out that solitary waves with various, in fact arbitrary, spatial structure and photon-like propagation properties exist. The {\it null} character of all non-linear vacuum solutions is established and extensively used further. Coordinate-free definitions are given to the important quantities {\it amplitude} and {\it phase}. The new quantity, named {\it scale factor}, is introduced and used as a criterion for availability of rotational component of propagation of some of the nonlinear, i.e. nonmaxwellian, vacuum solutions. The group structure properties of the nonlinear vacuum solutions are analyzed in some detail, showing explicitly the connection of the vacuum solutions with some complex valued functions. Connection-curvature interpretations are given and a special attention is paid to the curvature interpretation of the intrinsic rotational (spin) properties of some of the nonlinear solutions. Several approaches to coordinate-free local description and computation of the integral spin momentum are considered. Finally, a large family of nonvacuum spatial soliton-like solutions is explicitly written down, and a procedure to get (3+1) versions of the known (1+1) soliton solutions is obtained.

Derivation of special relativity classically relies on the use of inertial reference frames separat-ing at a constant velocity, combined with the constancy of the speed of light as a founding defi-nition of simultaneity. Einstein´s original method for derivation provides the general frame-work in which special relativity is often taught. While this approach is intuitive for students, it is nevertheless confusing in some aspects and lacks clarity in certain steps. Hence, special relativity is very often questioned with paradoxes and apparent mathematical inconsistencies. In most in-stances, these paradoxes are set up under improper mathematical frameworks, or in systems which are not invariant under coordinate transformations. Here, Minskowski space and special relativity are fully derived from an invariant tensor statement valid in any reference frame, with the simultaneity condition imposed on the 4-dimensional null vector. This approach derives Minkowski space with a very limited number of steps in tensor space without the need of in-troducing separating inertial reference frames. From it, the main features of special relativity for bradyon and tachyon trajectories are derived. This solution is particularly adequate for students of special and general relativity, as it is simple, straightforward and mathematically consistent.

The author's concept  and  postulate  of  what constitutes  the  whole  of  'the  Cosmos'  is:  "All  of  the  Infinite  'Absolute  Space' -'Time' Continuum".

Within this  multi-dimensionally  infinite  'Absolute  Space' -'Time'  Continuum'  there  is containment  of Finite Universes,  which  might  have  some varying  physics  laws  and  likely some  similar physics  laws  and  realities  as were  found  in  our  Universe.  The Universes  can have  varying  shapes,  however if  they  have  a 'cyclical'  existence,  then  their  likely  shapes should  be roughly  spherical,  or  ellipsoidal,  or  related  thereto.  These  Universes,  which  are within the  'Absolute  Space' -'Time' Continuum',  can be continually  evolving  and  their sizes  can  likely  pulsate,  which  means,  expand  and  contract  within  certain  limits.  These limits  can change  from  one  'Universe  Cycle' to  the  next.  The limits  of  expansion  depend on the  degree  of  occurrence of 'photon  decay'  activities,  which  leads  to  'mass'  increase and  on  the  value  of  a 'Maintenance'  Factor,  which  uses  the 'Gravitational  Inflow' mechanism.  This mechanism provides  that  there  is  no energy  loss  of  the  vortex  entities due  to  the  slight  friction,  which  occurs  there  where their  orderly  irrotational  flows  border onto  the  disorderly  'Brownian'  motion of the  cosmic  'Background  Radiation'.  This mechanism  finds  cause  in  the  phenomenon  of  'self-organization'  and  'self-defense' which  is  displayed  by 'closed  vortex  rings'.

'Absolute  Space'  is  everywhere,  also  within the  Universes,  which  are suspended therein.  Within  a  given  Universe, 'Absolute  Space' acts  as  a  'background'  therein,
against  which 'Time'  and  'Coordinates'  in  that  particular  Universe  can be  measured  or compared.

Universes  can  also  be 'super-positioned'  upon  each other, just as  well as  being next  (although  separate)  to  each other. 'Super-positioned'  means  that  one  Universe  is located  within  another  Universe  with the  understanding  that  all  Universes  are  suspended in  'Absolute  Space'.

Field theory in de Sitter space admits a one-parameter family of vacua determined by a superselection parameter alpha. Of these vacua, the Euclidean vacuum uniquely extrapolates to the vacuum of flat Minkowski space. States which resemble the alpha-vacua can be constructed as excitations above the Euclidean vacuum. Such states have modes alpha(k) which decay faster that k(1-d)/2. Fields in such

This paper contributes a new class of games called spacetime games with perfect information. In spacetime games, the agents make decisions at various positions in Minkowski spacetime. Spacetime games can be seen as the least common denominator of strategic games on the one hand, and dynamic games with perfect information on the other hand. Indeed, strategic games correspond to a configuration with only spacelike-separated decisions (&quot;different rooms&quot;). Dynamic games with perfect information, on the other hand, correspond to timelike-separated decisions (&quot;in turn&quot;). We show how to compute the strategic form and reduced strategic form of spacetime games. As a consequence, many existing solution concepts, such as Nash equilibria, rationalizability, individual rationality, etc, apply naturally to spacetime games. We introduce a canonical injection of the class of spacetime games with perfect information into the class of games in extensive form with imperfect informa...

Background and prehistory:
The French mathematician Henri Poincaré offered a statement known as “Poincaré’s conjecture” without a proof. It states that any 4-dimensional ball is equivalent to 3-dimensional Euclidean space topologically: a continuous mapping exists so that it maps the former ball into the latter space one-to-one.
At first glance, it seems to be too paradoxical for the following mismatches: the former is 4-dimensional and as if “closed” unlike the latter, 3-dimensional and as if “open” according to common sense. So, any mapping seemed to be necessarily discrete to be able to overcome those mismatches, and being discrete impies for the conjecture to be false.
Anyway, nobody managed neither to prove nor to reject rigorously the conjecture about one century. It was included even in the Millennium Prize Problems by the Clay Mathematics Institute.
It was proved by Grigory Perelman in 2003 using the concept of information.
Physical interpretation in terms of special relativity:
One may notice that the 4-ball is almost equivalent topologically to the “imaginary domain” of Minkowski space in the following sense of “almost”: that “half” of Minkowski space is equivalent topologically to the unfolding of a 4-ball. Then, the conjecture means the topological equivalence of the physical 3-space and its model in special relativity. In turn, that topological equivalence means their equivalence as to causality physically. So, Perelman has proved the adequacy of Minkowski space as a model of the physical 3-dimensional space rigorously. Of course, all experiments confirm the same empirically, but not mathematically as he did. 
An idea of another proof of the conjecture based on that physical interpretation:
Topologically seen, the problem turns out to be reformulated so: one needs a proof of the topological equivalence of a 4-ball and its unfolding by 3-balls (what the “half” of Minkowski space is, topologically).
If one adds a complementary, second unfolding to link both ends of the first unfolding, the problem would be resolved: 4-ball would be equivalent to two 3-spaces topologically. Two 3-spaces are equivalent to a single one as follows: one divides a 3-space into two parts by a certain plane (that plane does not belong to any of them). Any part is equivalent topologically to a 3-space for any open neighborhood is transformed into an open one by the mapping of each part (excluding the boundary of the plane) into the complete 3-space.
That idea is linked to the original proof of Perelman by the concept of information. It means that any bit of information interpreted physically conserves causality. In other words, the choice of any of both states of a bit (e.g. designated as “0” and “1” recorded in a cell) does not violate causality (the cell, either “0” or “1”, or both “0” and “1” are equivalent to each other topologically and to a 3-space).

We show that the kappa-deformed Poincaré quantum algebra proposed for particle physics has the structure of a Hopf algebra bicrossproduct U(so (1, 3)) T. The algebra is a semidirect product of the classical Lorentz group so(1,3) acting in a formed way on the momentum sector T. The novel feature is that the coalgebra is also semidirect, with a backreaction of

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system A (n) of observables “up to n loops ” where A (0) is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green’s functions.

We study the properties of an ultracold Fermi gas loaded in an optical square lattice and subjected to an external and classical non-Abelian gauge field. We show that this system can be exploited as an optical analogue of relativistic quantum electrodynamics, offering a remarkable route to access the exotic properties of massless Dirac fermions with cold atoms experiments. In particular we show that the underlying Minkowski space-time can also be modified, reaching anisotropic regimes where a remarkable anomalous quantum Hall effect and a squeezed Landau vacuum could be observed.

We discuss the possible breakdown of Lorentz invariance—at distances greater than the Planck length—from both the theoretical and the phenomenological point of view. The theoretical tool to deal with such a problem is provided by a deformation of the Minkowski metric, with parameters dependent on the energy of the physical system considered. Such a deformed metric realizes, for any interaction,