Weyl, “To understand nature, start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations . . .”

9-6-extra

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Gauge transformations appear of primarily descriptive nature only if we consider them in their function as changes of local (in the mathematical sense) changes of trivializations. In this function they are comparable to the transformations of the coordinates in a differentiable manifold, which also seem to have a purely “descriptive” function. But the coordinate changes stand in close relation to (local) diffeomorphisms. Therefore the postulate of coordinate independence of natural laws, or of the Lagrangian density, can and is being restated in terms of diffeomorphism invariance in general relativity. Similarly, the local changes of trivializations may be read as local descriptions.

The question as to whether or not the automorphisms express crucial physical properties  has nothing to do with the specific gauge nature of the groups, but hinges on the more overarching question of physical adequateness and physical content of the theory. The question of whether or why gauge symmetries can express physical content is not much different from the Kretschmann question of whether or why coordinate invariance of the laws, respectively coordinate covariance description of a physical theory, can have physical content. In the latter case the answer to the question has been dealt with in the philosophy of physics literature in great detail. Weyl’s answer is contained in his thoughts on the distinction of physical and mathematical automorphisms.

Let us shed a side-glance at gravitational gauge theories not taken into account by Weyl. In Einstein-Cartan gravity, which later turned out to be equivalent to Kibble-Sciama gravity, the localized rotational degrees of freedom lead to a conserved spin current and a non-symmetric energy tensor. This is a structurally pleasing effect, fitting roughly into the Noether charge paradigm, although with a peculiar “crossover” of the two Noether currents and the currents feeding the dynamical equations, inherited from Einstein gravity and Cartan’s identification of translational curvature with torsion. The rotational current, spin, feeds the dynamical equation of translational curvature; the translational current, energy-momentum, feeds the rotational curvature in the (generalized) Einstein equation. It may acquire physical relevance only if energy densities surpass the order of magnitude 1038 times the density of neutron stars. By this reason the current cannot yet be considered a physically striking effect. It may turn into one, if gravitational fields corresponding to extremely high energy densities acquire empirical relevance. For the time being, the rotational current can safely be neglected, Einstein-Cartan gravity reduces effectively to Einstein gravity, and Weyl’s argument for the symmetry of the energy-momentum tensor remains the most “striking consequence” in the sense of  rotational degrees of freedom.

On the other hand, the translational degrees of freedom give a more direct expression for the Noether currents of energy-momentum than the diffeomorphisms. The physical consequences for the diffeomorphism degrees of freedom reduce to the invariance constraint for the Lagrangian density for Einstein gravity considered as a special case of the Einstein-Cartan theory (with effectively vanishing spin). Besides these minor shifts, it may be more interesting to realize that the approach of Kibble and Sciama agreed nicely with Weyl’s methodological remark that for understanding nature we better “start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations . . . ”. This describes quite well what Sciama and Kibble did. They started to explore the consequences of localizing (in the physical sense) the translational and rotational degrees of freedom of special relativity. Their theory was built around the generalized automorphism group arising from localizing the Poincaré group.

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Category Theoretic (Mono-)/Sources

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A source is a pair (A,(fi)i∈I>) consisting of an object A and a family of morphisms fi : A → Ai with domain A, indexed by some class I. A is called the domain of the source and the family (Ai)i∈I is called the codomain of the source.

(1) Whenever convenient we use more concise notations, such as (A, fi)I, (A, fi) or fi

(A →fi Ai)I.

(2)  The indexing class I of a source (A,fi)I may be a proper class, a nonempty set, or the empty set. In case I = ∅, the source is determined by A. In case I ≠ ∅, the source is determined by the family (fi)I.

(3)  Sources indexed by the empty set are called empty sources and are denoted by (A,∅). Whenever convenient, objects may be regarded as empty sources.

(4)  Sources that are indexed by a set are called set-indexed or small.

(5)  Sources that are indexed by the set {1, . . . , n} are called n-sources and are denoted by (A, (f1, . . . , fn)). Whenever convenient, morphisms f : A → B may be regarded as 1-sources (A,f).

(6)  There are properties of sources that depend heavily on the fact that (fi)I is a family, i.e., an indexed collection (e.g., the property of being a product). There are other properties of sources (A,fi), depending on the domain A and the associated class {fi|i ∈ I} only (e.g., the property of being a mono-source). In order to avoid a clumsy distinction between indexed and non-indexed sources, we will sometimes regard classes as families (indexed by themselves via the corresponding identity function). Hence for any object A and any class S of morphisms with domain A, the pair (A,S) will be considered as a source. A particularly useful example is the total source (A,SA), where SA is the class of all morphisms with domain A.

If S = (A →fi Ai) I is a source and, for each i∈I, Si = (Aigij Aij) Ji is a source, then the source

(Si) ◦ S = (A →gij ◦ fi Aij) i ∈ I, j ∈ Ji

is called the composite of S and the family (Si)I.

(1) For a source S = (A → Ai)I and a morphism f : B → A we use the notation

S ◦ f = ( B →fi ◦f Ai)I .
(2) The composition of morphisms can be regarded as a special case of the composition of sources.
MONO-SOURCES

A source S = (A,fi)I is called a mono-source provided that it can be cancelled from the left, i.e., provided that for any pair B →r ←s A of morphisms the equation S ◦ r = S ◦ s (i.e., fi ◦ r = fi ◦ s for each i ∈ I) implies r = s.

PROPOSITION

(1) Representable functors preserve mono-sources (i.e., if G : A → Set is a representable functor and S is a mono-source in A, then GS is a mono-source in Set).

(2) Faithful functors reflect mono-sources (i.e., if G : A → B is a faithful functor, S = (A,fi) is a source in A, and GS = (GA,Gfi) is a mono-source in B, then S is a mono-source in A).

Proof:

(1). If a functor preserves mono-sources, then, clearly, so does every functor that is naturally isomorphic to it. Thus it suffices to show that each mono-source (B →fi Bi)I is sent by each hom-functor hom(A, −) : A → Set into a point-separating source:

(hom(A,B) →hom(A,fi) (hom(A,Bi))I

But this is immediate from the definition of mono-source.

(2). Let G and S be as described. If B →r ←s A is a pair of A-morphisms with S ◦ r = S ◦ s, then GS ◦ Gr = G(S ◦ r) = G(S ◦s) = GS ◦ Gs. Since GS is a mono-source, this implies Gr = Gs. Since G is faithful, this gives r = s.

COROLLARY

In a construct (A,U) every point-separating source is a mono-source. The converse holds whenever U is representable.

PROPOSITION

Let T = (Si) ◦ S be a composite of sources.

(1) If S and all Si are mono-sources, then so is T .

(2) If T is a mono-source, then so is S.

PROPOSITION

Let (A,fi)I be a source.

(1) If (A,fj)J is a mono-source for some J ⊆ I, then so is (A,fi)I.

(2) If fj is a monomorphism for some j ∈ I, then (A, fi)I is a mono-source.

A mono-source S is called extremal provided that whenever S = S ◦ e for some epimorphism e, then e must be an isomorphism.

PROPOSITION

(1) If a composite source (Si) ◦ S is an extremal mono-source, then so is S.

(2) If S ◦ f is an extremal mono-source, then f is an extremal monomorphism.

PROPOSITION

Let (A, fi)I be a source.

(1) If (A,fj)J is an extremal mono-source for some J ⊆ I, then so is (A, fi)I.

(2) If fj is an extremal monomorphism for some j ∈ I, then (A, fi)I is an extremal mono-source.

The concept of source allows a simple description of coseparators: namely, A is a coseparator if and only if, for any object B, the source (B,hom(B,A)) is a mono-source. This suggests the following definition:

An object A is called an extremal coseparator provided that for any object B the source (B, hom(B, A)) is an extremal mono-source.

Quantum Geometrodynamics and Emergence of Time in Quantum Gravity

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It is clear that, like quantum geometrodynamics, the functional integral approach makes fundamental use of a manifold. This means not just that it uses mathematical continua, such as the real numbers (to represent the values of coordinates, or physical quantities); it also postulates a 4-dimensional manifold M as an ‘arena for physical events’. However, its treatment of this manifold is very different from the treatment of spacetime in general relativity in so far as it has a Euclidean, not Lorentzian metric (which, apart from anything else, makes the use of the word ‘event’ distinctly problematic). Also, we may wish to make a summation over different such manifolds, it is in general necessary to consider complex metrics in the functional integral (so that the ‘distance squared’ between two spacetime points can be a complex number), whereas classical general relativity uses only real metrics.

Thus one might think that the manifold (or manifolds!) does not (do not) deserve the name ‘spacetime’. But what is in a name?! Let us in any case now ask how spacetime as understood in present-day physics could emerge from the above use of Riemannian manifolds M, perhaps taken together with other theoretical structures.

In particular: if we choose to specify the boundary conditions using the no-boundary proposal, this means that we take only those saddle-points of the action as contributors (to the semi-classical approximation of the wave function) that correspond to solutions of the Einstein field equations on a compact manifold M with a single boundary Σ and that induce the given values h and φ0 on Σ.

In this way, the question of whether the wave function defined by the functional integral is well approximated by this semi-classical approximation (and thus whether it predicts classical spacetime) turns out to be a question of choosing a contour of integration C in the space of complex spacetime metrics. For the approximation to be valid, we must be able to distort the contour C into a steepest-descents contour that passes through one or more of these stationary points and elsewhere follows a contour along which |e−I| decreases as rapidly as possible away from these stationary points. The wave function is then given by:

Ψ[h, φ0, Σ] ≈ ∑p e−Ip/ ̄h

where Ip are the stationary points of the action through which the contour passes, corresponding to classical solutions of the field equations satisfying the given boundary conditions. Although in general the integral defining the wave function will have many saddle-points, typically there is only a small number of saddle-points making the dominant contribution to the path integral.

For generic boundary conditions, no real Euclidean solutions to the classical Einstein field equations exist. Instead we have complex classical solutions, with a complex action. This accords with the account of the emergence of time via the semiclassical limit in quantum geometrodynamics.

On the Emergence of Time in Quantum Gravity

Gauge Theory of Arbitrage, or Financial Markets Resembling Screening in Electrodynamics

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When a mispricing appears in a market, market speculators and arbitrageurs rectify the mistake by obtaining a profit from it. In the case of profitable fluctuations they move into profitable assets, leaving comparably less profitable ones. This affects prices in such a way that all assets of similar risk become equally attractive, i.e. the speculators restore the equilibrium. If this process occurs infinitely rapidly, then the market corrects the mispricing instantly and current prices fully reflect all relevant information. In this case one says that the market is efficient. However, clearly it is an idealization and does not hold for small enough times.

The general picture, sketched above, of the restoration of equilibrium in financial markets resembles screening in electrodynamics. Indeed, in the case of electrodynamics, negative charges move into the region of the positive electric field, positive charges get out of the region and thus screen the field. Comparing this with the financial market we can say that a local virtual arbitrage opportunity with a positive excess return plays a role of the positive electric field, speculators in the long position behave as negative charges, whilst the speculators in the short position behave as positive ones. Movements of positive and negative charges screen out a profitable fluctuation and restore the equilibrium so that there is no arbitrage opportunity any more, i.e. the speculators have eliminated the arbitrage opportunity.

The analogy is apparently superficial, but it is not. The analogy emerges naturally in the framework of the Gauge Theory of Arbitrage (GTA). The theory treats a calculation of net present values and asset buying and selling as a parallel transport of money in some curved space, and interpret the interest rate, exchange rates and prices of asset as proper connection components. This structure is exactly equivalent to the geometrical structure underlying the electrodynamics where the components of the vector-potential are connection components responsible for the parallel transport of the charges. The components of the corresponding curvature tensors are the electromagnetic field in the case of electrodynamics and the excess rate of return in case of GTA. The presence of uncertainty is equivalent to the introduction of noise in the electrodynamics, i.e. quantization of the theory. It allows one to map the theory of the capital market onto the theory of quantized gauge field interacting with matter (money flow) fields. The gauge transformations of the matter field correspond to a change of the par value of the asset units which effect is eliminated by a gauge tuning of the prices and rates. Free quantum gauge field dynamics (in the absence of money flows) is described by a geometrical random walk for the assets prices with the log-normal probability distribution. In general case the consideration maps the capital market onto Quantum Electrodynamics where the price walks are affected by money flows.

Electrodynamical model of quasi-efficient financial market

Techno-Commercial Singularity: Decelerator / Diagram.

H/T Antinomia Imediata

If the Cathedral is actually efficient, the more it happens, the less it happens. Decelerator.

  1. taxation: this deviates resources from capital and buries them into the consumption of the tax-receivers (namely the Cathedral bureaucracy). trash and shit.
  2. regulation: there are various ways this could work, insofar as regulation is very inventive. but the main pattern has to do with deviating capital from the most rentable (i.e., (self-re)productive) investments, into those that are most likely to become un-recyclable trash, at least in the long run.
  3. politicization: this deviates brain-power from technological producing theories into, well, bullshit research departments, especially through politicization of academic funding of hard sciences.
  4. protectionism: since this protects technical developments from properly feeding back into the commercial cycle, it breaks the link between technical advantage and capital accumulation, leading lots of resources into stupid gadgetry.

all these being forms of fucking up the incentive structures that allow the accelerative cycle to be. in diagram form:unnamed (2)

 

 

Esotericism. Note Quote.

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So, here is some theosophy. Feels nostalgic, for I was a member at the Adyar Society, and these were the lingua franca then….

To the spiritual eagle eye of the seer and the prophet of every race, Ariadne’s thread stretches beyond that “historic period” without break or flaw, surely and steadily, into the very night of time; and the hand which holds it is too mighty to drop it, or even let it break. — The Secret Doctrine 2:67

This thread of esotericism stretches further still into the dawntime of the human race, where “Truth, high-seated upon its rock of adamant, is alone eternal and supreme” (Isis Unveiled 1:v). Where is this truth, this loom of esoteric history, and what the pattern of its fabric? In the home of the Brotherhood stands this loom, whose warp is the ancient threads of initiation held in occult tension by the sacrifice of adepts, and whose woof is woven century by century as each national unit spins the luminous threads of esotericism in its Mystery centers.

Profane history reveals scarcely anything of consecutive value, except insofar as the relics of this mystic pageantry all point to an identic motif. To “have a consecutive and full history of our race from its incipient stage down to the present times,” archaic records must needs be sought. By such alone could one trace even in faint outline the ancient pattern. Access to these, however, is the privilege of the adept alone, for such “knowledge is only for the highest Initiates, who do not take their students into their confidence” (Secret Doctrine 2:437-8). Nevertheless, we have received a priceless gift: the benefit of evidence gathered by those who have penetrated the veils of the adyta and have had the compassion to return and share with us a guarded portion of their vision. Study of their findings may at first reveal only a broken pattern, but if faithfully pursued such study will point with unmistakable clarity to one universal source of truth.

From Central Asia whose lands comprised a vast territory, including the Desert of Gobi or Shamo, the mountains of Tien Shan and Kuen Lun, the regions of Baluchistan, Afghanistan, Persia, and Turkestan, went forth bands of emigrants, in large part glowing with determination to conquer, to subdue, and many were the battles waged in those early days. A primal cause, yet unrecognized by the populace, was the urge of the Brotherhood aided by karma to carry the light of the Mysteries into other lands, to spread the ancient wisdom far and wide on the surface of the earth:

Not one people alone inhabited and built up these civilizations of Central Asia. They were recurrent waves of our present Fifth Root-Race. . . . each one of such civilizations being in its turn a cradle out of which grew children-colonies sent forth to carry light and initiation to what were then barbarous and uncultivated parts of the world, such as what is now Europe, what is now China, what is now Siberia, what is now India.– Studies in Occult Philosophy

To Bharata-varsha or India went forth the Aryas or “worthy ones,” a band of emigrants who founded a civilization and a culture as yet unrivaled in esoteric history, whose ramifications of spiritual influence extended to Egypt, Asia Minor, and Europe. Another band moved westward to Egypt, the “gift of the Nile” as Herodotus called it, and mixing with the aboriginal stock peopled its valleys. From this union sprang a princely civilization, the glory of which remains after thousands of centuries, the more so as the influence of its Mysteries spread far and wide as conquering nation after conquering nation became captive to the interior grandeur of Egypt. Persia, Babylonia, Judea and Crete, Greece and Rome, all trace their spiritual inspiration to the Egyptian and early Aryan cultures. Furthermore, so immense in esoteric power were these primeval civilizations, that there are records which show Egyptian priests — Initiates — journeying in a North-Westerly direction, by land, via what became later the Straits of Gibraltar; turning North and travelling through the future Phoenician settlements of Southern Gaul; then still further North, until reaching Carnac (Morbihan) they turned to the West again and arrived, still travelling by land, on the North-Western promontory of the New Continent [the British Isles].

What was the object of their long journey? And how far back must we place the date of such visits? The archaic records show the Initiates of the Second Sub-race of the Aryan family moving from one land to the other for the purpose of supervising the building of menhirs and dolmens, of colossal Zodiacs in stone, and places of sepulchre to serve as receptacles for the ashes of generations to come.– Secret Doctrine 2:750

What was the mainspring of these civilizations but the Mystery-teachings — teachings which penetrated into the very thought-life of nations, perhaps unknown of source, unrecognized by the multitudes as esoteric? Nonetheless they constituted the inspiration of the artist in his search for divinity, the intuition of the poet in his yearning for truth, and the resounding harmony of the musician as he sought the music of the spheres. It is no idle phrase to say that all things of spiritual, intellectual, and artistic value were born root and seed from the Sanctuary.

What are the stone and papyrus of Egypt but witness to knowledge of ancient truths long forgotten? The scenes of the weighing of the heart against the feather of truth in the papyri of Pert Em Hru — “Coming forth into Day,” commonly known as the Book of the Dead — portray in symbol and allegory what actually took place in the secret chambers of the initiation-pyramids. Living testimonial is the Great Pyramid of Khufu or Cheops, which H. P. Blavatsky hints more than once may go back at least 75,000 years BC, if not more (see Secret Doctrine 2:432, 750).

What of the Druids and their ancient ceremonies under oak and myrtle, with their stone monuments so oriented that the beams of the rising sun touched the brow of the candidate as he rose from his couch of initiation “clothed with the sun,” literally aflame with solar glory? Whence the training of their candidates in three degrees, a training which demanded absolute moral purity, spiritual vigor, and profound understanding of truth?

What about Persia and its long line of Zoroasters, within whose mystic seven-chambered centers truths of great intellectual and spiritual value were taught to neophytes who underwent the traditional discipline of the Mysteries? Were the Magi born from any other source than the archaic mother of occultism? What of the Orphic Mysteries, whose austere discipline and esoteric content may have had a stronger impact on Greek culture than the Eleusinian Mysteries, so popular for centuries? Do not the teachings of Orpheus indicate an Eastern origin reminiscent of those from India’s asramas or mystery-temples? Did not Pythagoras and Plato likewise travel India-wards and bring back to their disciples the identic pattern of esotericism?

Thus one might go on, with the Norse and Germanic mysticism, the Hindu and Chinese philosophies, the Greek and Roman ceremonial — all weavers of one pattern in one universal motif, a motif applicable to all times, to all nations, because capable of infinite variations. To grasp the inner purport of one Mystery school is to perceive the sacred identity of all of them — not in detail of cultural and ethnic interpretation, but in esoteric essentials.

What therefore is the test of truth? One basic requirement is universality: has it been taught by all those who have been “clothed with the central sun” of initiation? Did Buddha Gautama instruct his disciples in the selfsame doctrine that Christ Jesus did? Did Sankaracharya teach the same esotericism that Pythagoras and Empedocles did? Were Zoroaster and Tsong-kha-pa born in their adepthood from the same womb of the initiation chamber as Apollonius of Tyana, Orpheus, or Lao-Tz\bu? Have Persia and Greece, China and ancient America, Iceland, Wales, and Babylonia all received a message which, stripped of outer vestments, is one in essentials? Assuredly, it is so, for such patterns have been woven on one loom — the ageless loom of truth.

In the words of G. de Purucker, Fundamentals of the Esoteric Philosophy, 

These Mystery schools are not a unique system but, based on the spiritual structure of the universe, they were established from the same motives of compassion that presided over the acts of the great actors of the primal drama, the opening acts of our manvantara. They copied, as it were in miniature, what took place in those primordial times, and what took place in actual life in the Hierarchy of Compassion on our earth, or that section, rather, of the Hierarchy of Compassion which we call the Great White Lodge.

One primeval humanity, many children-colonies; one Mystery teaching, many Mystery schools; one archaic pattern, many variations of color and texture as each nation contributes the woof of its national Mysteries. Three are the variants of motif as seen from the present:

(1) the original unveiling of truth to infant humanity by divine instructors working in consonance with the spiritual planetary of our earth who, through the early millennia, successfully gathered together the choice few into a center of esoteric light – the great Brotherhood;

(2) the secondary unveiling springing directly as the fruit of the first: the spiritualizing influences uninterruptedly sent forth by the Brotherhood, and more specifically energized at cyclic intervals by their disciples, the great world teachers; and

(3) the third unveiling born as the progeny of (1) and (2): the penetration of truth into human life through the Mystery schools, the centers of esoteric discipline, in whose inner chambers initiation of the “elect” alone takes place, but in whose outer courts the world-at-large may seek entrance to learn fundamental verities so that life may be ennobled and death accepted as naturally as is sleep. Thus is the pattern of esotericism woven century by century on the loom of truth.