Automorphisms. Note Quote.

A group automorphism is an isomorphism from a group to itself. If  is a finite multiplicative group, an automorphism of  can be described as a way of rewriting its multiplication table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity  can be written as shown above, which means that the map defined by

is an automorphism of .

Looking at classical geometry and mechanics, Weyl followed Newton and Helmholtz in considering congruence as the basic relation which lay at the heart of the “art of measuring” by the handling of that “sort of bodies we call rigid”. He explained how the local congruence relations established by the comparison of rigid bodies can be generalized and abstracted to congruences of the whole space. In this respect Weyl followed an empiricist approach to classical physical geometry, based on a theoretical extension of the material practice with rigid bodies and their motions. Even the mathematical abstraction to mappings of the whole space carried the mark of their empirical origin and was restricted to the group of proper congruences (orientation preserving isometries of Euclidean space, generated by the translations and rotations) denoted by him as ∆⁺. This group seems to express “an intrinsic structure of space itself; a structure stamped by space upon all the inhabitants of space”.

But already on the earlier level of physical knowledge, so Weyl argued, the mathematical automorphisms of space were larger than ∆. Even if one sees “with Newton, in congruence the one and only basic concept of geometry from which all others derive”, the group Γ of automorphisms in the mathematical sense turns out to be constituted by the similarities.

The structural condition for an automorphism C ∈ Γ of classical congruence geometry is that any pair (v₁,v₂) of congruent geometric configurations is transformed into another pair (v₁^*,v₂^*) of congruent configurations (v_j^* = C(v_j), j = 1,2). For evaluating this property Weyl introduced the following diagram:

Because of the condition for automorphisms just mentioned the maps C T C^-1 and C^-1TC belong to ∆⁺ whenever T does. By this argument he showed that the mathematical automorphism group Γ is the normalizer of the congruences ∆⁺ in the group of bijective mappings of Euclidean space.

More generally, it also explains the reason for his characterization of generalized similarities in his analysis of the problem of space in the early 1920s. In 1918 he translated the relationship between physical equivalences as congruences to the mathematical automorphisms as the similarities/normalizer of the congruences from classical geometry to special relativity (Minkowski space) and “localized” them (in the sense of physics), i.e., he transferred the structural relationship to the infinitesimal neighbourhoods of the differentiable manifold characterizing spacetime (in more recent language, to the tangent spaces) and developed what later would be called Weylian manifolds, a generalization of Riemannian geometry. In his discussion of the problem of space he generalized the same relationship even further by allowing any (closed) sub-group of the general linear group as a candidate for characterizing generalized congruences at every point.

Moreover, Weyl argued that the enlargement of the physico-geometrical automorphisms of classical geometry (proper congruences) by the mathematical automorphisms (similarities) sheds light on Kant’s riddle of the “incongruous counterparts”. Weyl presented it as the question: Why are “incongruous counterparts” like the left and right hands intrinsically indiscernible, although they cannot be transformed into another by a proper motion? From his point of view the intrinsic indiscernibility could be characterized by the mathematical automorphisms Γ. Of course, the congruences ∆ including the reflections are part of the latter, ∆ ⊂ Γ; this implies indiscernibility between “left and right” as a special case. In this way Kant’s riddle was solved by a Leibnizian type of argument. Weyl very cautiously indicated a philosophical implication of this observation:

And he (Kant) is inclined to think that only transcendental idealism is able to solve this riddle. No doubt, the meaning of congruence and similarity is founded in spatial intuition. Kant seems to aim at some subtler point. But just this point is one which can be completely clarified by general concepts, namely by subsuming it under the general and typical group-theoretic situation explained before . . . .

Weyl stopped here without discussing the relationship between group theoretical methods and the “subtler point” Kant aimed at more explicitly. But we may read this remark as an indication that he considered his reflections on automorphism groups as a contribution to the transcendental analysis of the conceptual constitution of modern science. In his book on Symmetry, he went a tiny step further. Still with the Weylian restraint regarding the discussion of philosophical principles he stated: “As far as I see all a priori statements in physics have their origin in symmetry” (126).

To prepare for the following, Weyl specified the subgroup ∆o ⊂ ∆ with all those transformations that fix one point (∆_o = O(3, R), the orthogonal group in 3 dimensions, R the field of real numbers). In passing he remarked:

In the four-dimensional world the Lorentz group takes the place of the orthogonal group. But here I shall restrict myself to the three-dimensional space, only occasionally pointing to the modifications, the inclusion of time into the four-dimensional world brings about.

Keeping this caveat in mind (restriction to three-dimensional space) Weyl characterized the “group of automorphisms of the physical world”, in the sense of classical physics (including quantum mechanics) by the combination (more technically, the semidirect product ̧) of translations and rotations, while the mathematical automorphisms arise from a normal extension:

– physical automorphisms ∆ ≅ R³ X| ∆_o with ∆_o ≅ O(3), respectively ∆ ≅ R⁴ X| ∆_o for the Lorentz group ∆_o ≅ O(1, 3),

– mathematical automorphisms Γ = R⁺ X ∆
(R⁺ the positive real numbers with multiplication).

In Weyl’s view the difference between mathematical and physical automorphisms established a fundamental distinction between mathematical geometry and physics.

Congruence, or physical equivalence, is a geometric concept, the meaning of which refers to the laws of physical phenomena; the congruence group ∆ is essentially the group of physical automorphisms. If we interpret geometry as an abstract science dealing with such relations and such relations only as can be logically defined in terms of the one concept of congruence, then the group of geometric automorphisms is the normalizer of ∆ and hence wider than ∆.

He considered this as a striking argument against what he considered to be the Cartesian program of a reductionist geometrization of physics (physics as the science of res extensa):

According to this conception, Descartes’s program of reducing physics to geometry would involve a vicious circle, and the fact that the group of geometric automorphisms is wider than that of physical automorphisms would show that such a reduction is actually impossible.” 

In this Weyl alluded to an illusion he himself had shared for a short time as a young scientist. After the creation of his gauge geometry in 1918 and the proposal of a geometrically unified field theory of electromagnetism and gravity he believed, for a short while, to have achieved a complete geometrization of physics.

He gave up this illusion in the middle of the 1920s under the impression of the rising quantum mechanics. In his own contribution to the new quantum mechanics groups and their linear representations played a crucial role. In this respect the mathematical automorphisms of geometry and the physical automorphisms “of Nature”, or more precisely the automorphisms of physical systems, moved even further apart, because now the physical automorphism started to take non-geometrical material degrees of freedom into account (phase symmetry of wave functions and, already earlier, the permutation symmetries of n-particle systems).

But already during the 19th century the physical automorphism group had acquired a far deeper aspect than that of the mobility of rigid bodies:

In physics we have to consider not only points but many types of physical quantities such as velocity, force, electromagnetic field strength, etc. . . .

All these quantities can be represented, relative to a Cartesian frame, by sets of numbers such that any orthogonal transformation T performed on the coordinates keeps the basic physical relations, the physical laws, invariant. Weyl accordingly stated:

All the laws of nature are invariant under the transformations thus induced by the group ∆. Thus physical relativity can be completely described by means of a group of transformations of space-points.

By this argumentation Weyl described a deep shift which ocurred in the late 19th century for the understanding of physics. He described it as an extension of the group of physical automorphisms. The laws of physics (“basic relations” in his more abstract terminology above) could no longer be directly characterized by the motion of rigid bodies because the physics of fields, in particular of electric and magnetic fields, had become central. In this context, the motions of material bodies lost their epistemological primary status and the physical automorphisms acquired a more abstract character, although they were still completely characterizable in geometric terms, by the full group of Euclidean isometries. The indistinguishability of left and right, observed already in clear terms by Kant, acquired the status of a physical symmetry in electromagnetism and in crystallography.

Weyl thus insisted that in classical physics the physical automorphisms could be characterized by the group ∆ of Euclidean isometries, larger than the physical congruences (proper motions) ∆⁺ but smaller than the mathe- matical automorphisms (similarities) Γ.

This view fitted well to insights which Weyl drew from recent developments in quantum physics. He insisted – differently to what he had thought in 1918 – on the consequence that “length is not relative but absolute” (Hs, p. 15). He argued that physical length measurements were no longer dependent on an arbitrary chosen unit, like in Euclidean geometry. An “absolute standard of length” could be fixed by the quantum mechanical laws of the atomic shell:

The atomic constants of charge and mass of the electron atomic constants and Planck’s quantum of action h, which enter the universal field laws of nature, fix an absolute standard of length, that through the wave lengths of spectral lines is made available for practical measurements.

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Catharsis

One of the ways in which Freud was able to reveal repressed unconscious representations in discourse was through a technique popularly known as “the talking cure”. Coined by “Anna O” – a patient of Josef Breuer, Freud’s family doctor – the talking cure was considered by Freud to be effective in the treatment of hysteria. The technique, not unlike the notion of “free association”, requires the patient to say out loud whatever comes to her/his mind no matter how insignificant or superficial it may seem. By encouraging the patient to concentrate on putting neurotic or psychotic experiences into words, the uninterrupted narrative flow allows the psychoanalyst to reconstruct the patient’s unconscious mind. Focusing on unconscious representations revealed during discourse, the psychoanalyst is able to perceive lapsus or other manifestations of the unconscious which may escape the patient’s field of perception. At the discretion of the psychoanalyst, these unconscious representations are in turn brought to the patient’s attention who will ideally be able to understand the root of an undesired behavior. According to Freud, becoming familiar with the exact nature of these neurotic and psychotic behaviors makes it possible for the patient to suppress them.

The success of the talking cure, also known as the cathartic method, depends on the patient’s ability to put thoughts into words through the free assembly of signifiers. Through extensive research on the connection between the spoken word and the idea it represented, Freud argued that the organization of words, as well as their subsequent verbalization, have a direct link not only with cognition, but also with kinesthetics. In their analysis of the case of “Anna O.” – a patient of Breuer suffering from acute hysteria – both Freud and Breuer recognized the therapeutic benefits of the cathartic method. During the course of her hysteria, the patient essentially repressed the anguish of her father’s death into the unconscious mind, the cathexis of which resurfaced as a series of somatic manifestations. The patient’s symptoms, ranging from partial paralysis to severe coughing, completely disappeared toward the final phases of her treatment, much to the surprise of Freud and Breuer. They later attributed the patient’s cure to her verbalized reenactment of emotionally charged scenes associated with her father’s death, in the same manner as Aristotle remarked on the soothing effects of catharsis.

Further elaborating on Freud’s relationship between thoughts and words, Lacan perceived the unconscious mind. as being comprised of individual signifiers. Combining Saussurian linguistics and Freudian psychoanalysis, Lacan’s perception of the unconscious mind expounded on the ‘word-presentations’ mentioned by Freud in The Ego and the Id. Whereas Freud conceived the unconscious mind as containing “thing-presentations” that could be verbalized in the conscious mind, only by their subsequent passage through the pre-conscious, Lacan demonstrated that these “thing-presentations” already behave like signifiers without first having to filter through the pre-conscious. Lacan points out that the unconscious is manifested not only in speech through unconscious lapsus, but also in dreams, qualified by Freud as “the via regia to the unconscious”.

Because dreams both contain verbal cues and take on characteristics of linguistic tropes such as metaphor and metonymy, Lacan reasons that the unconscious must be structured like a language. To support this theory, he likens metaphor and metonymy to two functions of Freud’s dream-work: condensation and displacement, respectively. According to Lacan, metaphor behaves like condensation in that a signifier belonging to a particular signifying chain can be substituted with a new signifier from a different signifying chain in order to be reassigned a new meaning. Thus, metaphor appears both in narration and in dreams when a signifier-word is attributed a meaning other than that which is normally associated with it. In this way, condensation acts as a censoring agent to protect the ego from images, drives or impulses that it has repressed. Closely related to metonymy, dreams can also be censored through displacement. Instead of compressing images, drives or impulses into a metaphor as is the case with condensation, displacement disguises unconscious representations by replacing a repressed signifier in a signifying chain with another signifier from the same chain. This implies that the signifier that has been replaced in the signifying chain is related to the new signifier, as is the case of metonymy which uses only one part of a thing to describe the whole thing.

It would appear that, like language, the unconscious is governed by the relationship between individual units, in much the same way that words are governed by the rules of grammar and tropes to create meaning. In this respect, not only are unconscious and conscious signifiers similar to one another, but Freud’s cathartic method further corroborates their equivalence. With the assistance of a psychoanalyst, the “talking cure” brings unconscious drives, impulses and the images they create to the conscious realm through psychic discharge, which in the context of psychoanalysis, takes on the form of verbalized discourse. Instead of remaining confined to the unconscious and surfacing in unexpected or undesired ways through psychotic or neurotic behaviors, unconscious cathexes are channeled into language which, as Freud pointed out in “Words and Things”, is closely related to somatic activity. If unconscious cathexes can be converted into speech instead of into debilitating behaviors, then the connection between elements of the unconscious and those of the conscious can be clearly established.

But, for Freud, art is (as is love) an attenuated and inhibited form of sexuality that has a “mildly intoxicating quality of feeling.” The full power of human affects is exhausted and satisfied only in sexuality, “the prototype of all happiness.” For Lacan, the deepest passions are not localized or limited to genital sexuality, but engage the entire corporeal being in many, unpredictable forms of jouissance. Art is a way into jouissance. By doing violence to its own structural and meaning-making properties, art bewilders, perplexes, shocks, or enraptures, causing a “resonating of the body” that the speaking being (‘parlêtre’) wants and enjoys, even at the price of pain or anxiety. It has techniques and ways of making interventions that psychoanalysis can perhaps adapt for producing an encounter in the analysand with his or her own wordless real. By contrast, though Freud praised art for preceding psychoanalysis in understanding our psychic constitution, he did not see it as having any kind of direct application or usefulness for analytic practice. For the late Lacan, psychoanalysis is no longer the Freudian “talking cure” but a search for new paths to accomplish a kind of tuning of the jouissance that underlies all thought and discourse.

Irrationality. Note Quote.

To mathematics it is unique, that two absolutely contrary opinions do not logically exclude each other but exist simultaneously while there seems to be no chance to pick out a false one and to establish a remaining truth. This case is realised by the philosophy and mathematics of the infinite. While transfinite set theory is impossible without different degrees of infinity, constructivists and intuitionists deny this notion without running into inconsistencies as is admitted by some of the foremost set theorists:

… the attitude of the (neo-)intuitionists that there do not exist altogether non-equivalent infinite sets is consistent, though almost suicidal for mathematics. [p. 62]

It would not be astonishing if in different axiomatic systems different results were obtained with respect to peculiarities of those systems. But set theorists on one side and constructivists and intuitionists on the other are certainly believing to address the same entities when speaking of “rational numbers” or of “irrational numbers”. In spite of that, the former are convinced that there are infinitely many more irrational numbers than rational numbers while the latter deny that:

Hence the continua of Weyl, Lebesgue, Lusin, etc. are denumerable … [p. 255]

This situation yields bewildering results:

Feferman and Levy showed that one cannot prove that there is any non-denumerable set of real numbers which can be well ordered. … Moreover, they also showed that the statement that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted. [p. 62]

Nevertheless, the great majority of mathematicians refuse to accept the thesis that Cantor’s ideas were but a pathological fancy. Though the foundations of set theory are still somewhat shaky. Most surprising and by no means to be expected of a pupil of Fraenkel’s is that Robinson states:

Infinite totalities do not exist in any sense of the word (i.e. either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. Nevertheless, we should act as if infinite totalities really existed. [3]

Does there exist a correct and an incorrect position? And, if so, who is right, who is wrong?

Following the advice of Fraenkel, namely to judge about the value and necessity of the basic axioms, in particular of the axiom of choice, by considering its consequences, in order to settle this question. These consequences will turn out to entail what, in an euphemistic way, by set theorists usually is called a “paradoxical result”, in order to avoid the term self-contradiction.

Apart from the well-ordering theorem some statements of quite different character – in particular geometrical statements – have been proved by means of the axiom of choice, which because of their paradoxical character induced some mathematicians to reject the axiom. Presumably the earliest statement of this kind is Hausdorff’s discovery that half of the sphere’s surface is congruent to a third of it. … It may surprise scholars working in the field … that even after more than half a century of utilising the axiom of choice and well-ordering theorem, a number of first-rate mathematicians (especially French) have not essentially changed their distrustful attitude.

Transfinite set theory arises from Cantor’s observation that the set of all irrational numbers has infinitely many more members than the set of all rational numbers. While the latter has the same cardinality χ₀ as the set N of all natural numbers n, the cardinality χ of the set of all irrational numbers is larger, χ = 2^χ₀. It is proven to be uncountable, i.e., any bijection with N can be excluded.

Comment on Purely Random Correlations of the Matrix, or Studying Noise in Neural Networks

In the presence of two-body interactions the many-body Hamiltonian matrix elements v^J_α,α′ of good total angular momentum J in the shell-model basis |α⟩ generated by the mean field, can be expressed as follows:

v^J_α,α′ = ∑^J’_ii’ c^Jαα’_J’ii’ g^J’_ii’ —– (4)

The summation runs over all combinations of the two-particle states |i⟩ coupled to the angular momentum J′ and connected by the two-body interaction g. The analogy of this structure to the one schematically captured by the eq. (2) is evident. g^J’_ii’ denote here the radial parts of the corresponding two-body matrix elements while c^Jαα’_J’ii’ globally represent elements of the angular momentum recoupling geometry. g^J’_ii’ are drawn from a Gaussian distribution while the geometry expressed by c^Jαα’_J’ii’ enters explicitly. This originates from the fact that a quasi-random coupling of individual spins results in the so-called geometric chaoticity and thus c^Jαα’ coefficients are also Gaussian distributed. In this case, these two (g^J’_ii’ and c) essentially random ingredients lead however to an order of magnitude larger separation of the ground state from the remaining states as compared to a pure Random Matrix Theory (RMT) limit. Due to more severe selection rules the effect of geometric chaoticity does not apply for J = 0. Consistently, the ground state energy gaps measured relative to the average level spacing characteristic for a given J is larger for J > 0 than for J = 0, and also J > 0 ground states are more orderly than those for J = 0, as it can be quantified in terms of the information entropy.

Interestingly, such reductions of dimensionality of the Hamiltonian matrix can also be seen locally in explicit calculations with realistic (non-random) nuclear interactions. A collective state, the one which turns out coherent with some operator representing physical external field, is always surrounded by a reduced density of states, i.e., it repells the other states. In all those cases, the global fluctuation characteristics remain however largely consistent with the corresponding version of the random matrix ensemble.

Recently, a broad arena of applicability of the random matrix theory opens in connection with the most complex systems known to exist in the universe. With no doubt, the most complex is the human’s brain and those phenomena that result from its activity. From the physics point of view the financial world, reflecting such an activity, is of particular interest because its characteristics are quantified directly in terms of numbers and a huge amount of electronically stored financial data is readily available. An access to a single brain activity is also possible by detecting the electric or magnetic fields generated by the neuronal currents. With the present day techniques of electro- or magnetoencephalography, in this way it is possible to generate the time series which resolve neuronal activity down to the scale of 1 ms.

One may debate over what is more complex, the human brain or the financial world, and there is no unique answer. It seems however to us that it is the financial world that is even more complex. After all, it involves the activity of many human brains and it seems even less predictable due to more frequent changes between different modes of action. Noise is of course owerwhelming in either of these systems, as it can be inferred from the structure of eigen-spectra of the correlation matrices taken across different space areas at the same time, or across different time intervals. There however always exist several well identifiable deviations, which, with help of reference to the universal characteristics of the random matrix theory, and with the methodology briefly reviewed above, can be classified as real correlations or collectivity. An easily identifiable gap between the corresponding eigenvalues of the correlation matrix and the bulk of its eigenspectrum plays the central role in this connection. The brain when responding to the sensory stimulations develops larger gaps than the brain at rest. The correlation matrix formalism in its most general asymmetric form allows to study also the time-delayed correlations, like the ones between the oposite hemispheres. The time-delay reflecting the maximum of correlation (time needed for an information to be transmitted between the different sensory areas in the brain is also associated with appearance of one significantly larger eigenvalue. Similar effects appear to govern formation of the heteropolymeric biomolecules. The ones that nature makes use of are separated by an energy gap from the purely random sequences.

 

Purely Random Correlations of the Matrix, or Studying Noise in Neural Networks

Expressed in the most general form, in essentially all the cases of practical interest, the n × n matrices W used to describe the complex system are by construction designed as

W = XY^T —– (1)

where X and Y denote the rectangular n × m matrices. Such, for instance, are the correlation matrices whose standard form corresponds to Y = X. In this case one thinks of n observations or cases, each represented by a m dimensional row vector x_i (y_i), (i = 1, …, n), and typically m is larger than n. In the limit of purely random correlations the matrix W is then said to be a Wishart matrix. The resulting density ρW(λ) of eigenvalues is here known analytically, with the limits (λ_min ≤ λ ≤ λ_max) prescribed by

λ^max_min = 1+1/Q±2 1/Q and Q = m/n ≥ 1.

The variance of the elements of x_i is here assumed unity.

The more general case, of X and Y different, results in asymmetric correlation matrices with complex eigenvalues λ. In this more general case a limiting distribution corresponding to purely random correlations seems not to be yet known analytically as a function of m/n. It indicates however that in the case of no correlations, quite generically, one may expect a largely uniform distribution of λ bound in an ellipse on the complex plane.

Further examples of matrices of similar structure, of great interest from the point of view of complexity, include the Hamiltonian matrices of strongly interacting quantum many body systems such as atomic nuclei. This holds true on the level of bound states where the problem is described by the Hermitian matrices, as well as for excitations embedded in the continuum. This later case can be formulated in terms of an open quantum system, which is represented by a complex non-Hermitian Hamiltonian matrix. Several neural network models also belong to this category of matrix structure. In this domain the reference is provided by the Gaussian (orthogonal, unitary, symplectic) ensembles of random matrices with the semi-circle law for the eigenvalue distribution. For the irreversible processes there exists their complex version with a special case, the so-called scattering ensemble, which accounts for S-matrix unitarity.

As it has already been expressed above, several variants of ensembles of the random matrices provide an appropriate and natural reference for quantifying various characteristics of complexity. The bulk of such characteristics is expected to be consistent with Random Matrix Theory (RMT), and in fact there exists strong evidence that it is. Once this is established, even more interesting are however deviations, especially those signaling emergence of synchronous or coherent patterns, i.e., the effects connected with the reduction of dimensionality. In the matrix terminology such patterns can thus be associated with a significantly reduced rank k (thus k ≪ n) of a leading component of W. A satisfactory structure of the matrix that would allow some coexistence of chaos or noise and of collectivity thus reads:

W = W_r + W_c —– (2)

Of course, in the absence of W_r, the second term (W_c) of W generates k nonzero eigenvalues, and all the remaining ones (n − k) constitute the zero modes. When W_r enters as a noise (random like matrix) correction, a trace of the above effect is expected to remain, i.e., k large eigenvalues and the bulk composed of n − k small eigenvalues whose distribution and fluctuations are consistent with an appropriate version of random matrix ensemble. One likely mechanism that may lead to such a segregation of eigenspectra is that m in eq. (1) is significantly smaller than n, or that the number of large components makes it effectively small on the level of large entries w of W. Such an effective reduction of m (M = m_eff) is then expressed by the following distribution P(w) of the large off-diagonal matrix elements in the case they are still generated by the random like processes

P(w) = (|w|^(M-1)/2K_(M-1)/2(|w|))/(2^(M-1)/2Γ(M/2)√π) —– (3)

where K stands for the modified Bessel function. Asymptotically, for large w, this leads to P(w) ∼ e(−|w|) |w|^M/2−1, and thus reflects an enhanced probability of appearence of a few large off-diagonal matrix elements as compared to a Gaussian distribution. As consistent with the central limit theorem the distribution quickly converges to a Gaussian with increasing M.

Based on several examples of natural complex dynamical systems, like the strongly interacting Fermi systems, the human brain and the financial markets, one could systematize evidence that such effects are indeed common to all the phenomena that intuitively can be qualified as complex.