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

# 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 (vj* = C(vj), 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.

# 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 vJα,α′ of good total angular momentum J in the shell-model basis |α⟩ generated by the mean field, can be expressed as follows:

vJα,α′ = ∑J’ii’ cJαα’J’ii’ gJ’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. gJ’ii’ denote here the radial parts of the corresponding two-body matrix elements while cJαα’J’ii’ globally represent elements of the angular momentum recoupling geometry. gJ’ii’ are drawn from a Gaussian distribution while the geometry expressed by cJαα’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 cJαα’ coefficients are also Gaussian distributed. In this case, these two (gJ’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.