Black Hole Entropy in terms of Mass. Note Quote.


If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

GD = G11/Ld = l911/Ld —– (1)

A Schwartzschild black hole of mass M has a radius

Rs ~ M(1/(D-3)) GD(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4GD —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ RsD-2 —– (4)


S ~ 1/GD (MGD)(D-2)/(D-3) ~ M(D-2)/(D-3) GD1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

Nmin(S) = MRs = M(MGD)1/D-3 = S —– (6)

We see that the value of Nmin in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = Nmin = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑3 of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T33 —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N2.

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N2T43 —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N23((E/N23))3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M2/N R —– (11)

Plugging (11) into (10)

S = N23(M2R/N23)3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

Capital As Power.


One has the Eric Fromm angle of consciousness as linear and directly proportional to exploitation as one of the strands of Marxian thinking, the non-linearity creeps up from epistemology on the technological side, with, something like, say Moore’s Law, where ascension of conscious thought is or could be likened to exponentials. Now, these exponentials are potent in ridding of the pronouns, as in the “I” having a compossibility with the “We”, for if these aren’t gotten rid of, there is asphyxiation in continuing with them, an effort, an energy expendable into the vestiges of waste, before Capitalism comes sweeping in over such deliberately pronounced islands of pronouns. This is where the sweep is of the “IT”. And this is emancipation of the highest order, where teleology would be replaced by Eschatology. Alienation would be replaced with emancipation. Teleology is alienating, whereas eschatology is emancipating. Agency would become un-agency. An emancipation from alienation, from being, into the arms of becoming, for the former is a mere snapshot of the illusory order, whereas the latter is a continuum of fluidity, the fluid dynamics of the deracinated from the illusory order. The “IT” is pure and brute materialism, the cosmic unfoldings beyond our understanding and importantly mirrored in on the terrestrial. “IT” is not to be realized. “It” is what engulfs us, kills us, and in the process emancipates us from alienation. “IT” is “Realism”, a philosophy without “we”, Capitalism’s excessive power. “IT” enslaves “us” to the point of us losing any identification. In a nutshell, theory of capital is a catalogue of heresies to be welcomed to set free from the vantage of an intention to emancipate economic thought from the etherealized spheres of choice and behaviors or from the paradigm of the disembodied minds.

Jonathan Nitzan and Shimshon Bichler‘s Capital as Power A Study of Order and Creorder

Bacteria’s Perception-Action Circle: Materiality of the Ontological. Thought of the Day 136.0


The unicellular organism has thin filaments protruding from its cell membrane, and in the absence of any stimuli, it simply wanders randomly around by changing between two characteristical movement patterns. One is performed by rotating the flagella counterclockwise. In that case, they form a bundle which pushes the cell forward along a curved path, a ‘run’ of random duration with these runs interchanging with ‘tumbles’ where the flagella shifts to clockwise rotation, making them work independently and hence moving the cell erratically around with small net displacement. The biased random walk now consists in the fact than in the presence of a chemical attractant, the runs happening to carry the cell closer to the attractant are extended, while runs in other directions are not. The sensation of the chemical attractant is performed temporally rather than spatially, because the cell moves too rapidly for concentration comparisons between its two ends to be possible. A chemical repellant in the environment gives rise to an analogous behavioral structure – now the biased random walk takes the cell away from the repellant. The bias saturates very quickly – which is what prevents the cell from continuing in a ‘false’ direction, because a higher concentration of attractant will now be needed to repeat the bias. The reception system has three parts, one detecting repellants such as leucin, the other detecting sugars, the third oxygen and oxygen-like substances.


The cell’s behavior forms a primitive, if full-fledged example of von Uexküll’s functional circle connecting specific perception signs and action signs. Functional circle behavior is thus no privilege for animals equipped with central nervous systems (CNS). Both types of signs involve categorization. First, the sensory receptors of the bacterium evidently are organized after categorization of certain biologically significant chemicals, while most chemicals that remain insignificant for the cell’s metabolism and survival are ignored. The self-preservation of metabolism and cell structure is hence the ultimate regulator which is supported by the perception-action cycles described. The categorization inherent in the very structure of the sensors is mirrored in the categorization of act types. Three act types are outlined: a null-action, composed of random running and tumbling, and two mirroring biased variants triggered by attractants and repellants, respectively. Moreover, a negative feed-back loop governed by quick satiation grants that the window of concentration shifts to which the cell is able to react appropriately is large – it so to speak calibrates the sensory system so that it does not remain blinded by one perception and does not keep moving the cell forward on in one selected direction. This adaptation of the system grants that it works in a large scale of different attractor/repellor concentrations. These simple signals at stake in the cell’s functional circle display an important property: at simple biological levels, the distinction between signs and perception vanish – that distinction is supposedly only relevant for higher CNS-based animals. Here, the signals are based on categorical perception – a perception which immediately categorizes the entity perceived and thus remains blind to internal differences within the category.

Pandemic e coli

The mechanism by which the cell identifies sugar, is partly identical to what goes on in human taste buds. Sensation of sugar gradients must, of course, differ from the consumption of it – while the latter, of course, destroys the sugar molecule, the former merely reads an ‘active site’ on the outside of the macromolecule. E . Coli – exactly like us – may be fooled by artificial sweeteners bearing the same ‘active site’ on their outer perimeter, even if being completely different chemicals (this is, of course, the secret behind such sweeteners, they are not sugars and hence do not enter the digestion process carrying the energy of carbohydrates). This implies that E . coli may be fooled. Bacteria may not lie, but a simpler process than lying (which presupposes two agents and the ability of being fooled) is, in fact, being fooled (presupposing, in turn, only one agent and an ambiguous environment). E . coli has the ability to categorize a series of sugars – but, by the same token, the ability to categorize a series of irrelevant substances along with them. On the one hand, the ability to recognize and categorize an object by a surface property only (due to the weak van der Waal-bonds and hydrogen bonds to the ‘active site’, in contrast to the strong covalent bonds holding the molecule together) facilitates perception economy and quick action adaptability. On the other hand, the economy involved in judging objects from their surface only has an unavoidable flip side: it involves the possibility of mistake, of being fooled by allowing impostors in your categorization. So in the perception-action circle of a bacterium, some of the self-regulatory stability of a metabolism involving categorized signal and action involvement with the surroundings form intercellular communication in multicellular organisms to reach out to complicated perception and communication in higher animals.

Fermi Surface Singularities

In ideal Fermi gases, the Fermi surface at p = pF = √2μm is the boundary in p-space between the occupied states (np = 1) at p2/2m < μ and empty states (np = 0) at p2/2m > μ. At this boundary (the surface in 3D momentum space) the energy is zero. What happens when the interaction between particles is introduced? Due to interaction the distribution function np of particles in the ground state is no longer exactly 1 or 0. However, it appears that the Fermi surface survives as the singularity in np. Such stability of the Fermi surface comes from a topological property of the one-particle Green’s function at imaginary frequency:

G-1 = iω – p2/2m + μ —– (1)

Let us for simplicity skip one spatial dimension pz so that the Fermi surface becomes the line in 2D momentum space (px,py); this does not change the co-dimension of zeroes which remains 1 = 3−2 = 2−1. The Green’s function has singularities lying on a closed line ω = 0, p2x + p2y = p2F in the 3D momentum-frequency space (ω,px,py). This is the line of the quantized vortex in the momentum space, since the phase Φ of the Green’s function G = |G|e changes by 2πN1 around the path embracing any element of this vortex line. In the considered case the phase winding number is N1 = 1. If we add the third momentum dimension pz the vortex line becomes the surface in the 4D momentum-frequency space (ω,px,py,pz) – the Fermi surface – but again the phase changes by 2π along any closed loop empracing the element of the 2D surface in the 4D momentum-frequency space.


Fermi surface is a topological object in momentum space – a vortex loop. When the chemical potential μ decreases the loop shrinks and disappears at μ < 0. The point μ = T = 0 marks the Lifshitz transition between the gapless ground state at μ > 0 to the fully gapped vacuum at μ < 0.

The winding number cannot change by continuous deformation of the Green’s function: the momentum-space vortex is robust toward any perturbation. Thus the singularity of the Green’s function on the Fermi surface is preserved, even when interaction between fermions is introduced. The invariant is the same for any space dimension, since the co-dimension remains 1.

The Green function is generally a matrix with spin indices. In addition, it may have the band indices (in the case of electrons in the periodic potential of crystals). In such a case the phase of the Green’s function becomes meaning-less; however, the topological property of the Green’s function remains robust. The general analysis demonstrates that topologically stable Fermi surfaces are described by the group Z of integers. The winding number N1 is expressed analytically in terms of the Green’s function:

N= tr ∮C dl/2πi G(μ,p) ∂lG-1(μ,p) —– (2)

Here the integral is taken over an arbitrary contour C around the momentum- space vortex, and tr is the trace over the spin, band and/or other indices.

The Fermi surface cannot be destroyed by small perturbations, since it is protected by topology and thus is robust to perturbations. But the Fermi surface can be removed by large perturbations in the processes which reproduces the processes occurring for the real-space counterpart of the Fermi surface – the loop of quantized vortex in superfluids and superconductors. The vortex ring can continuously shrink to a point and then disappear, or continuously expand and leave the momentum space. The first scenario occurs when one continuously changes the chemical potential from the positive to the negative value: at μ < 0 there is no vortex loop in momentum space and the ground state (vacuum) is fully gapped. The point μ = 0 marks the quantum phase transition – the Lifshitz transition – at which the topology of the energy spectrum changes. At this transition the symmetry of the ground state does not changes. The second scenario of the quantum phase transition to the fully gapped states occurs when the inverse mass 1/m in (1) crosses zero.

Similar Lifshitz transitions from the fully gapped state to the state with the Fermi surface may occur in superfluids and superconductors. This happens, for example, when the superfluid velocity crosses the Landau critical velocity. The symmetry of the order parameter does not change across such a quantum phase transition. In the non-superconduting states, the transition from the gapless to gapped state is the metal-insulator transition.


The Lifshitz transitions involving the vortex lines in p-space may occur be- tween the gapless states. They are accompanied by the change of the topology of the Fermi surface itself. The simplest example of such a phase transition discussed in terms of the vortex lines is provided by the reconnection of the vortex lines.



Dynamics of Point Particles: Orthogonality and Proportionality


Let γ be a smooth, future-directed, timelike curve with unit tangent field ξa in our background spacetime (M, gab). We suppose that some massive point particle O has (the image of) this curve as its worldline. Further, let p be a point on the image of γ and let λa be a vector at p. Then there is a natural decomposition of λa into components proportional to, and orthogonal to, ξa:

λa = (λbξba + (λa −(λbξba) —– (1)

Here, the first part of the sum is proportional to ξa, whereas the second one is orthogonal to ξa.

These are standardly interpreted, respectively, as the “temporal” and “spatial” components of λa relative to ξa (or relative to O). In particular, the three-dimensional vector space of vectors at p orthogonal to ξa is interpreted as the “infinitesimal” simultaneity slice of O at p. If we introduce the tangent and orthogonal projection operators

kab = ξa ξb —– (2)

hab = gab − ξa ξb —– (3)

then the decomposition can be expressed in the form

λa = kab λb + hab λb —– (4)

We can think of kab and hab as the relative temporal and spatial metrics determined by ξa. They are symmetric and satisfy

kabkbc = kac —– (5)

habhbc = hac —– (6)

Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas. For example, suppose that the worldline of a second particle O′ also passes through p and that its four-velocity at p is ξ′a. (Since ξa and ξ′a are both future-directed, they are co-oriented; i.e., ξa ξ′a > 0.) We compute the speed of O′ as determined by O. To do so, we take the spatial magnitude of ξ′a relative to O and divide by its temporal magnitude relative to O:

v = speed of O′ relative to O = ∥hab ξ′b∥ / ∥kab ξ′b∥ —– (7)

For any vector μa, ∥μa∥ is (μaμa)1/2 if μ is causal, and it is (−μaμa)1/2 otherwise.

We have from equations 2, 3, 5 and 6

∥kab ξ′b∥ = (kab ξ′b kac ξ′c)1/2 = (kbc ξ′bξ′c)1/2 = (ξ′bξb)


∥hab ξ′b∥ = (−hab ξ′b hac ξ′c)1/2 = (−hbc ξ′bξ′c)1/2 = ((ξ′bξb)2 − 1)1/2


v = ((ξ’bξb)2 − 1)1/2 / (ξ′bξb) < 1 —– (8)

Thus, as measured by O, no massive particle can ever attain the maximal speed 1. We note that equation (8) implies that

(ξ′bξb) = 1/√(1 – v2) —– (9)

It is a basic fact of relativistic life that there is associated with every point particle, at every event on its worldline, a four-momentum (or energy-momentum) vector Pa that is tangent to its worldline there. The length ∥Pa∥ of this vector is what we would otherwise call the mass (or inertial mass or rest mass) of the particle. So, in particular, if Pa is timelike, we can write it in the form Pa =mξa, where m = ∥Pa∥ > 0 and ξa is the four-velocity of the particle. No such decomposition is possible when Pa is null and m = ∥Pa∥ = 0.

Suppose a particle O with positive mass has four-velocity ξa at a point, and another particle O′ has four-momentum Pa there. The latter can either be a particle with positive mass or mass 0. We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose Pa in terms of ξa. By equations (4) and (2), we have

Pa = (Pbξb) ξa + habPb —– (10)

the first part of the sum is the energy component, while the second is the three-momentum. The energy relative to O is the coefficient in the first term: E = Pbξb. If O′ has positive mass and Pa = mξ′a, this yields, by equation (9),

E = m (ξ′bξb) = m/√(1 − v2) —– (11)

(If we had not chosen units in which c = 1, the numerator in the final expression would have been mc2 and the denominator √(1 − (v2/c2)). The three−momentum relative to O is the second term habPb in the decomposition of Pa, i.e., the component of Pa orthogonal to ξa. It follows from equations (8) and (9) that it has magnitude

p = ∥hab mξ′b∥ = m((ξ′bξb)2 − 1)1/2 = mv/√(1 − v2) —– (12)

Interpretive principle asserts that the worldlines of free particles with positive mass are the images of timelike geodesics. It can be thought of as a relativistic version of Newton’s first law of motion. Now we consider acceleration and a relativistic version of the second law. Once again, let γ : I → M be a smooth, future-directed, timelike curve with unit tangent field ξa. Just as we understand ξa to be the four-velocity field of a massive point particle (that has the image of γ as its worldline), so we understand ξnnξa – the directional derivative of ξa in the direction ξa – to be its four-acceleration field (or just acceleration) field). The four-acceleration vector at any point is orthogonal to ξa. (This is, since ξannξa) = 1/2 ξnnaξa) = 1/2 ξnn (1) = 0). The magnitude ∥ξnnξa∥ of the four-acceleration vector at a point is just what we would otherwise describe as the curvature of γ there. It is a measure of the rate at which γ “changes direction.” (And γ is a geodesic precisely if its curvature vanishes everywhere).

The notion of spacetime acceleration requires attention. Consider an example. Suppose you decide to end it all and jump off the tower. What would your acceleration history be like during your final moments? One is accustomed in such cases to think in terms of acceleration relative to the earth. So one would say that you undergo acceleration between the time of your jump and your calamitous arrival. But on the present account, that description has things backwards. Between jump and arrival, you are not accelerating. You are in a state of free fall and moving (approximately) along a spacetime geodesic. But before the jump, and after the arrival, you are accelerating. The floor of the observation deck, and then later the sidewalk, push you away from a geodesic path. The all-important idea here is that we are incorporating the “gravitational field” into the geometric structure of spacetime, and particles traverse geodesics iff they are acted on by no forces “except gravity.”

The acceleration of our massive point particle – i.e., its deviation from a geodesic trajectory – is determined by the forces acting on it (other than “gravity”). If it has mass m, and if the vector field Fa on I represents the vector sum of the various (non-gravitational) forces acting on it, then the particle’s four-acceleration ξnnξa satisfies

Fa = mξnnξa —– (13)

This is Newton’s second law of motion. Consider an example. Electromagnetic fields are represented by smooth, anti-symmetric fields Fab. If a particle with mass m > 0, charge q, and four-velocity field ξa is present, the force exerted by the field on the particle at a point is given by qFabξb. If we use this expression for the left side of equation (13), we arrive at the Lorentz law of motion for charged particles in the presence of an electromagnetic field:

qFabξb = mξbbξa —– (14)

This equation makes geometric sense. The acceleration field on the right is orthogonal to ξa. But so is the force field on the left, since ξa(Fabξb) = ξaξbFab = ξaξbF(ab), and F(ab) = 0 by the anti-symmetry of Fab.

Iain Hamilton Grant’s Schelling in Opposition to Fichte. Note Quote.


The stated villain of Philosophies of Nature is not Hegelianism but rather ‘neo-Fichteanism’. It is Grant’s ‘Philosophies of Nature After Schelling‘, which takes up the issue of graduating Schelling to escape the accoutrements of Kantian and Fichtean narrow transcendentalism. Grant frees Schelling from the grips of narrow minded inertness and mechanicality in nature that Kant and Fichte had presented nature with. This idea is the Deleuzean influence on Grant. Manuel De Landa makes a vociferous case in this regard. According to De Landa, the inertness of matter was rubbished by Deleuze in the way that Deleuze sought for a morphogenesis of form thereby launching a new kind of materialism. This is the anti-essentialist position of Deleuze. Essentialism says that matter and energy are inert, they do not have any morphogenetic capabilities. They cannot give rise to new forms on their own. Disciplines like complexity theory, non-linear dynamics do give matter its autonomy over inertness, its capabilities in terms of charge. But its account of the relationship between Fichte and Schelling actually obscures the rich meaning of speculation in Hegel and after. Grant quite accurately recalls that Schelling confronted Fichte’s identification of the ‘not I’ with passive nature – the consequence of identifying all free activity with the ‘I’ alone. For Grant, that which Jacobi termed ‘speculative egotism’ becomes the nightmare of modern philosophy and of technological modernity at large. The ecological concern is never quite made explicit in Philosophies of Nature. Yet Grant’s introduction to Schelling’s On the World Soul helps to contextualise the meaning of his ‘geology of morals’.

What we miss from Grant’s critique of Fichte is the manner by which the corrective, positive characterisation of nature proceeds from Schelling’s confirmation of Fichte’s rendering of the fact of consciousness (Tatsache) into the act of consciousness (Tathandlung). Schelling, as a consequence, becomes singularly critical of contemplative speculation, since activity now implies working on nature and thereby changing it – along with it, we might say – rather than either simply observing it or even experimenting upon it.

In fact, Grant reads Schelling only in opposition to Fichte, with drastic consequences for his speculative realism: the post-Fichtean element of Schelling’s naturephilosophy allows for the new sense of speculation he will share with Hegel – even though they will indeed turn this against Kant and Fichte. Without this account, we are left with the older, contemplative understanding of metaphysical speculation, which leads to a certain methodologism in Grant’s study. Hence, ‘the principle method of naturephilosophy consists in “unconditioning” the phenomena’. Relatedly, Meillassoux defines the ‘speculative’ as ‘every type of thinking’ – not acting, – ‘that claims to be able to access some form of absolute’.

In direct contrast to this approach, the collective ‘system programme’ of Hegel, Schelling and Hölderlin was not a programme for thinking alone. Their revolutionised sense of speculation, from contemplation of the stars to reform of the worldly, is overlooked by today’s speculative realism – a philosophy that, ‘refuses to interrogate reality through human (linguistic, cultural or political) mediations of it’. We recall that Kant similarly could not extend his Critique to speculative reason precisely on account of his contemplative determination of pure reason (in terms of the hierarchical gap between reason and the understanding). Grant’s ‘geology of morals’ does not oppose ‘Kanto-Fichtean philosophy’, as he has it, but rather remains structurally within the sphere of Kant’s pre-political metaphysics.

The Chernobyl Herbarium


Some images in Anaïs Tondeur’s Chernobyl Herbarium are the explosions of light. Others are softly glowing, breathing with fragility and precariousness. The explosive imprints are, in effect, reminiscent of volcanic eruptions at night, hot lava spewing from the depths of the earth. Even assuming it is not an actual trace of radiation (which the specimens in the herbarium have received from the isotopes of cesium-137 and strontium-90 mixed with the soil of the exclusion zone) that comes through and shines forth from the plants’ contact with photosensitive paper, the resulting works of art cannot help but send us back to a space and time outside the frame, wherein this Linum usitatissimum germinated, grew, and blossomed.

The images are the visible records of an invisible calamity, tracked across the threshold of sight by the power of art. The literal translation from Greek of the technique used here, photogram, is a line of light. Not a photograph, the writing of light, but a photogram, its line captured on photosensitive paper, upon which the object is placed. In writing, a line is already too idealized, too heavy with meaning, overburdened with sense, nearly immaterial. In a photograph, light’s imprint is further removed from the being that emitted or reflected it than in a photogram, where, absent the camera, the line can be itself, can trace itself outside the system of coded significations and machinic mediations. The grammé of a photogram imposes itself from up close. Touching… It endures: etched, engraved, engrained, the energy it transported both reflected (or refracted) and absorbed. Much like radiation, indifferently imbibed by whatever and whoever is on its path – the soil, buildings, plants, animals, humans – yet uncontainable in any single entity whose time-frame it invariably overflows. Through her aesthetic practice, Tondeur detonates, releases the explosions of light trapped in plants, its lines dispersed, crisscrossing photograms every which way. She liberates luminescent traces without violence, avoiding the repetition of the first, invisible event of Chernobyl and, at the same time, capturing something of it. Release and preservation; preservation and release: by the grace of art.

The Chernobyl Herbarium




Let us introduce the concept of space using the notion of reflexive action (or reflex action) between two things. Intuitively, a thing x acts on another thing y if the presence of x disturbs the history of y. Events in the real world seem to happen in such a way that it takes some time for the action of x to propagate up to y. This fact can be used to construct a relational theory of space à la Leibniz, that is, by taking space as a set of equitemporal things. It is necessary then to define the relation of simultaneity between states of things.

Let x and y be two things with histories h(xτ) and h(yτ), respectively, and let us suppose that the action of x on y starts at τx0. The history of y will be modified starting from τy0. The proper times are still not related but we can introduce the reflex action to define the notion of simultaneity. The action of y on x, started at τy0, will modify x from τx1 on. The relation “the action of x on y is reflected to x” is the reflex action. Historically, Galileo introduced the reflection of a light pulse on a mirror to measure the speed of light. With this relation we will define the concept of simultaneity of events that happen on different basic things.


Besides we have a second important fact: observation and experiment suggest that gravitation, whose source is energy, is a universal interaction, carried by the gravitational field.

Let us now state the above hypothesis axiomatically.

Axiom 1 (Universal interaction): Any pair of basic things interact. This extremely strong axiom states not only that there exist no completely isolated things but that all things are interconnected.

This universal interconnection of things should not be confused with “universal interconnection” claimed by several mystical schools. The present interconnection is possible only through physical agents, with no mystical content. It is possible to model two noninteracting things in Minkowski space assuming they are accelerated during an infinite proper time. It is easy to see that an infinite energy is necessary to keep a constant acceleration, so the model does not represent real things, with limited energy supply.

Now consider the time interval (τx1 − τx0). Special Relativity suggests that it is nonzero, since any action propagates with a finite speed. We then state

Axiom 2 (Finite speed axiom): Given two different and separated basic things x and y, such as in the above figure, there exists a minimum positive bound for the interval (τx1 − τx0) defined by the reflex action.

Now we can define Simultaneity as τy0 is simultaneous with τx1/2 =Df (1/2)(τx1 + τx0)

The local times on x and y can be synchronized by the simultaneity relation. However, as we know from General Relativity, the simultaneity relation is transitive only in special reference frames called synchronous, thus prompting us to include the following axiom:

Axiom 3 (Synchronizability): Given a set of separated basic things {xi} there is an assignment of proper times τi such that the relation of simultaneity is transitive.

With this axiom, the simultaneity relation is an equivalence relation. Now we can define a first approximation to physical space, which is the ontic space as the equivalence class of states defined by the relation of simultaneity on the set of things is the ontic space EO.

The notion of simultaneity allows the analysis of the notion of clock. A thing y ∈ Θ is a clock for the thing x if there exists an injective function ψ : SL(y) → SL(x), such that τ < τ′ ⇒ ψ(τ) < ψ(τ′). i.e.: the proper time of the clock grows in the same way as the time of things. The name Universal time applies to the proper time of a reference thing that is also a clock. From this we see that “universal time” is frame dependent in agreement with the results of Special Relativity.

US Stock Market Interaction Network as Learned by the Boltzmann Machine


Price formation on a financial market is a complex problem: It reflects opinion of investors about true value of the asset in question, policies of the producers, external regulation and many other factors. Given the big number of factors influencing price, many of which unknown to us, describing price formation essentially requires probabilistic approaches. In the last decades, synergy of methods from various scientific areas has opened new horizons in understanding the mechanisms that underlie related problems. One of the popular approaches is to consider a financial market as a complex system, where not only a great number of constituents plays crucial role but also non-trivial interaction properties between them. For example, related interdisciplinary studies of complex financial systems have revealed their enhanced sensitivity to fluctuations and external factors near critical events with overall change of internal structure. This can be complemented by the research devoted to equilibrium and non-equilibrium phase transitions.

In general, statistical modeling of the state space of a complex system requires writing down the probability distribution over this space using real data. In a simple version of modeling, the probability of an observable configuration (state of a system) described by a vector of variables s can be given in the exponential form

p(s) = Z−1 exp {−βH(s)} —– (1)

where H is the Hamiltonian of a system, β is inverse temperature (further β ≡ 1 is assumed) and Z is a statistical sum. Physical meaning of the model’s components depends on the context and, for instance, in the case of financial systems, s can represent a vector of stock returns and H can be interpreted as the inverse utility function. Generally, H has parameters defined by its series expansion in s. Basing on the maximum entropy principle, expansion up to the quadratic terms is usually used, leading to the pairwise interaction models. In the equilibrium case, the Hamiltonian has form

H(s) = −hTs − sTJs —– (2)

where h is a vector of size N of external fields and J is a symmetric N × N matrix of couplings (T denotes transpose). The energy-based models represented by (1) play essential role not only in statistical physics but also in neuroscience (models of neural networks) and machine learning (generative models, also known as Boltzmann machines). Given topological similarities between neural and financial networks, these systems can be considered as examples of complex adaptive systems, which are characterized by the adaptation ability to changing environment, trying to stay in equilibrium with it. From this point of view, market structural properties, e.g. clustering and networks, play important role for modeling of the distribution of stock prices. Adaptation (or learning) in these systems implies change of the parameters of H as financial and economic systems evolve. Using statistical inference for the model’s parameters, the main goal is to have a model capable of reproducing the same statistical observables given time series for a particular historical period. In the pairwise case, the objective is to have

⟨sidata = ⟨simodel —– (3a)

⟨sisjdata = ⟨sisjmodel —– (3b)

where angular brackets denote statistical averaging over time. Having specified general mathematical model, one can also discuss similarities between financial and infinite- range magnetic systems in terms of phenomena related, e.g. extensivity, order parameters and phase transitions, etc. These features can be captured even in the simplified case, when si is a binary variable taking only two discrete values. Effect of the mapping to a binarized system, when the values si = +1 and si = −1 correspond to profit and loss respectively. In this case, diagonal elements of the coupling matrix, Jii, are zero because s2i = 1 terms do not contribute to the Hamiltonian….

US stock market interaction network as learned by the Boltzmann Machine

The Locus of Renormalization. Note Quote.


Since symmetries and the related conservation properties have a major role in physics, it is interesting to consider the paradigmatic case where symmetry changes are at the core of the analysis: critical transitions. In these state transitions, “something is not preserved”. In general, this is expressed by the fact that some symmetries are broken or new ones are obtained after the transition (symmetry changes, corresponding to state changes). At the transition, typically, there is the passage to a new “coherence structure” (a non-trivial scale symmetry); mathematically, this is described by the non-analyticity of the pertinent formal development. Consider the classical para-ferromagnetic transition: the system goes from a disordered state to sudden common orientation of spins, up to the complete ordered state of a unique orientation. Or percolation, often based on the formation of fractal structures, that is the iteration of a statistically invariant motif. Similarly for the formation of a snow flake . . . . In all these circumstances, a “new physical object of observation” is formed. Most of the current analyses deal with transitions at equilibrium; the less studied and more challenging case of far form equilibrium critical transitions may require new mathematical tools, or variants of the powerful renormalization methods. These methods change the pertinent object, yet they are based on symmetries and conservation properties such as energy or other invariants. That is, one obtains a new object, yet not necessarily new observables for the theoretical analysis. Another key mathematical aspect of renormalization is that it analyzes point-wise transitions, that is, mathematically, the physical transition is seen as happening in an isolated mathematical point (isolated with respect to the interval topology, or the topology induced by the usual measurement and the associated metrics).

One can say in full generality that a mathematical frame completely handles the determination of the object it describes as long as no strong enough singularity (i.e. relevant infinity or divergences) shows up to break this very mathematical determination. In classical statistical fields (at criticality) and in quantum field theories this leads to the necessity of using renormalization methods. The point of these methods is that when it is impossible to handle mathematically all the interaction of the system in a direct manner (because they lead to infinite quantities and therefore to no relevant account of the situation), one can still analyze parts of the interactions in a systematic manner, typically within arbitrary scale intervals. This allows us to exhibit a symmetry between partial sets of “interactions”, when the arbitrary scales are taken as a parameter.

In this situation, the intelligibility still has an “upward” flavor since renormalization is based on the stability of the equational determination when one considers a part of the interactions occurring in the system. Now, the “locus of the objectivity” is not in the description of the parts but in the stability of the equational determination when taking more and more interactions into account. This is true for critical phenomena, where the parts, atoms for example, can be objectivized outside the system and have a characteristic scale. In general, though, only scale invariance matters and the contingent choice of a fundamental (atomic) scale is irrelevant. Even worse, in quantum fields theories, the parts are not really separable from the whole (this would mean to separate an electron from the field it generates) and there is no relevant elementary scale which would allow ONE to get rid of the infinities (and again this would be quite arbitrary, since the objectivity needs the inter-scale relationship).

In short, even in physics there are situations where the whole is not the sum of the parts because the parts cannot be summed on (this is not specific to quantum fields and is also relevant for classical fields, in principle). In these situations, the intelligibility is obtained by the scale symmetry which is why fundamental scale choices are arbitrary with respect to this phenomena. This choice of the object of quantitative and objective analysis is at the core of the scientific enterprise: looking only at molecules as the only pertinent observable of life is worse than reductionist, it is against the history of physics and its audacious unifications and invention of new observables, scale invariances and even conceptual frames.

As for criticality in biology, there exists substantial empirical evidence that living organisms undergo critical transitions. These are mostly analyzed as limit situations, either never really reached by an organism or as occasional point-wise transitions. Or also, as researchers nicely claim in specific analysis: a biological system, a cell genetic regulatory networks, brain and brain slices …are “poised at criticality”. In other words, critical state transitions happen continually.

Thus, as for the pertinent observables, the phenotypes, we propose to understand evolutionary trajectories as cascades of critical transitions, thus of symmetry changes. In this perspective, one cannot pre-give, nor formally pre-define, the phase space for the biological dynamics, in contrast to what has been done for the profound mathematical frame for physics. This does not forbid a scientific analysis of life. This may just be given in different terms.

As for evolution, there is no possible equational entailment nor a causal structure of determination derived from such entailment, as in physics. The point is that these are better understood and correlated, since the work of Noether and Weyl in the last century, as symmetries in the intended equations, where they express the underlying invariants and invariant preserving transformations. No theoretical symmetries, no equations, thus no laws and no entailed causes allow the mathematical deduction of biological trajectories in pre-given phase spaces – at least not in the deep and strong sense established by the physico-mathematical theories. Observe that the robust, clear, powerful physico-mathematical sense of entailing law has been permeating all sciences, including societal ones, economics among others. If we are correct, this permeating physico-mathematical sense of entailing law must be given up for unentailed diachronic evolution in biology, in economic evolution, and cultural evolution.

As a fundamental example of symmetry change, observe that mitosis yields different proteome distributions, differences in DNA or DNA expressions, in membranes or organelles: the symmetries are not preserved. In a multi-cellular organism, each mitosis asymmetrically reconstructs a new coherent “Kantian whole”, in the sense of the physics of critical transitions: a new tissue matrix, new collagen structure, new cell-to-cell connections . . . . And we undergo millions of mitosis each minute. More, this is not “noise”: this is variability, which yields diversity, which is at the core of evolution and even of stability of an organism or an ecosystem. Organisms and ecosystems are structurally stable, also because they are Kantian wholes that permanently and non-identically reconstruct themselves: they do it in an always different, thus adaptive, way. They change the coherence structure, thus its symmetries. This reconstruction is thus random, but also not random, as it heavily depends on constraints, such as the proteins types imposed by the DNA, the relative geometric distribution of cells in embryogenesis, interactions in an organism, in a niche, but also on the opposite of constraints, the autonomy of Kantian wholes.

In the interaction with the ecosystem, the evolutionary trajectory of an organism is characterized by the co-constitution of new interfaces, i.e. new functions and organs that are the proper observables for the Darwinian analysis. And the change of a (major) function induces a change in the global Kantian whole as a coherence structure, that is it changes the internal symmetries: the fish with the new bladder will swim differently, its heart-vascular system will relevantly change ….

Organisms transform the ecosystem while transforming themselves and they can stand/do it because they have an internal preserved universe. Its stability is maintained also by slightly, yet constantly changing internal symmetries. The notion of extended criticality in biology focuses on the dynamics of symmetry changes and provides an insight into the permanent, ontogenetic and evolutionary adaptability, as long as these changes are compatible with the co-constituted Kantian whole and the ecosystem. As we said, autonomy is integrated in and regulated by constraints, with an organism itself and of an organism within an ecosystem. Autonomy makes no sense without constraints and constraints apply to an autonomous Kantian whole. So constraints shape autonomy, which in turn modifies constraints, within the margin of viability, i.e. within the limits of the interval of extended criticality. The extended critical transition proper to the biological dynamics does not allow one to prestate the symmetries and the correlated phase space.

Consider, say, a microbial ecosystem in a human. It has some 150 different microbial species in the intestinal tract. Each person’s ecosystem is unique, and tends largely to be restored following antibiotic treatment. Each of these microbes is a Kantian whole, and in ways we do not understand yet, the “community” in the intestines co-creates their worlds together, co-creating the niches by which each and all achieve, with the surrounding human tissue, a task closure that is “always” sustained even if it may change by immigration of new microbial species into the community and extinction of old species in the community. With such community membership turnover, or community assembly, the phase space of the system is undergoing continual and open ended changes. Moreover, given the rate of mutation in microbial populations, it is very likely that these microbial communities are also co-evolving with one another on a rapid time scale. Again, the phase space is continually changing as are the symmetries.

Can one have a complete description of actual and potential biological niches? If so, the description seems to be incompressible, in the sense that any linguistic description may require new names and meanings for the new unprestable functions, where functions and their names make only sense in the newly co-constructed biological and historical (linguistic) environment. Even for existing niches, short descriptions are given from a specific perspective (they are very epistemic), looking at a purpose, say. One finds out a feature in a niche, because you observe that if it goes away the intended organisms dies. In other terms, niches are compared by differences: one may not be able to prove that two niches are identical or equivalent (in supporting life), but one may show that two niches are different. Once more, there are no symmetries organizing over time these spaces and their internal relations. Mathematically, no symmetry (groups) nor (partial-) order (semigroups) organize the phase spaces of phenotypes, in contrast to physical phase spaces.

Finally, here is one of the many logical challenges posed by evolution: the circularity of the definition of niches is more than the circularity in the definitions. The “in the definitions” circularity concerns the quantities (or quantitative distributions) of given observables. Typically, a numerical function defined by recursion or by impredicative tools yields a circularity in the definition and poses no mathematical nor logical problems, in contemporary logic (this is so also for recursive definitions of metabolic cycles in biology). Similarly, a river flow, which shapes its own border, presents technical difficulties for a careful equational description of its dynamics, but no mathematical nor logical impossibility: one has to optimize a highly non linear and large action/reaction system, yielding a dynamically constructed geodetic, the river path, in perfectly known phase spaces (momentum and space or energy and time, say, as pertinent observables and variables).

The circularity “of the definitions” applies, instead, when it is impossible to prestate the phase space, so the very novel interaction (including the “boundary conditions” in the niche and the biological dynamics) co-defines new observables. The circularity then radically differs from the one in the definition, since it is at the meta-theoretical (meta-linguistic) level: which are the observables and variables to put in the equations? It is not just within prestatable yet circular equations within the theory (ordinary recursion and extended non – linear dynamics), but in the ever changing observables, the phenotypes and the biological functions in a circularly co-specified niche. Mathematically and logically, no law entails the evolution of the biosphere.