Are Categories Similar to Sets? A Folly, if General Relativity and Quantum Mechanics Think So.

The fundamental importance of the path integral suggests that it might be enlightening to simplify things somewhat by stripping away the knot observable K and studying only the bare partition functions of the theory, considered over arbitrary spacetimes. That is, consider the path integral

Z(M) = ∫ DA e (i ∫_M S(A) —– (1)

where M is an arbitrary closed 3d manifold, that is, compact and without boundary, and S[A] is the Chern-Simons action. Immediately one is struck by the fact that, since the action is topological, the number Z(M) associated to M should be a topological invariant of M. This is a remarkably efficient way to produce topological invariants.

Poincaré Conjecture: If M is a closed 3-manifold, whose fundamental group π₁(M), and all of whose homology groups H_i(M) are equal to those of S³, then M is homeomorphic to S³.

One therefore appreciates the simplicity of the quantum field theory approach to topological invariants, which runs as follows.

1. Endow the space with extra geometric structure in the form of a connection (alternatively a field, a section of a line bundle, an embedding map into spacetime)
2. Compute a number from this manifold-with-connection (the action)
3. Sum over all connections.

This may be viewed as an extension of the general principle in mathematics that one should classify structures by the various kinds of extra structure that can live on them. Indeed, the Chern-Simons Lagrangian was originally introduced in mathematics in precisely this way. Chern-Weil theory provides access to the cohomology groups (that is, topological invariants) of a manifold M by introducing an arbitrary connection A on M, and then associating to A a closed form f(A) (for instance, via the Chern-Simons Lagrangian), whose cohomology class is, remarkably, independent of the original arbitrary choice of connection A. Quantum field theory takes this approach to the extreme by being far more ambitious; it associates to a connection A the actual numerical value of the action (usually obtained by integration over M) – this number certainly depends on the connection, but field theory atones for this by summing over all connections.

Quantum field theory is however, in its path integral manifestation, far more than a mere machine for computing numbers associated with manifolds. There is dynamics involved, for the natural purpose of path integrals is not to calculate bare partition functions such as equation (1), but rather to express the probability amplitude for a given field configuration to evolve into another. Thus one considers a 3d manifold M (spacetime) with boundary components Σ₁ and Σ₂ (space), and considers M as the evolution of space from its initial configuration Σ₁ to its final configuration Σ₂:

This is known mathematically as a cobordism from Σ₁ to Σ₂. To a 2d closed manifold Σ we associate the space of fields A(Σ) living on Σ. A physical state Ψ corresponds to a functional on this space of fields. This is the Schrödinger picture of quantum field theory: if A ∈ A(Σ), then Ψ(A) represents the probability that the state known as Ψ will be found in the field A. Such a state evolves with time due to the dynamics of the theory; Ψ(A) → Ψ(A, t). The space of states has a natural basis, which consists of the delta functionals  – these are the states satisfying ⟨Â|Â′⟩ = δ(A − A′). Any arbitrary state Ψ may be expressed as a superposition of these basis states. The path integral instructs us how to compute the time evolution of states, by first expanding them in the  basis, and then specifying that the amplitude for a system in the state Â₁ on the space Σ₁ to be found in the state Â₂ on the space Σ₂ is given by:

〈Â2|U|Â1〉= ∫A | ∑2 = A2 A | ∑1 = A1 DA e i S[A] —– (2)

This equation is the fundamental formula of quantum field theory: ‘Perform a weighted sum over all possible fields (connections) living on spacetime that restrict to A₁ and A2 on Σ₁ and Σ₂ respectively’. This formula constructs the time evolution operator U associated to the cobordism M.

In this way we see that, at the very heart of quantum mechanics and quantum field theory, is a formula which associates to every space-like manifold Σ a Hilbert space of fields A(Σ), and to every cobordism M from Σ₁ to Σ₂ a time evolution operator U(M) : Σ₁ – Σ₂. To specify a quantum field theory is nothing more than to give rules for constructing the Hilbert spaces A(Σ) and the rules (correlation functions) for calculating the time evolution operators U(M). This is precisely the statement that a quantum field theory is a functor from the cobordism category nCob to the category of Hilbert spaces Hilb.

A category C consists of a collection of objects, a collection of arrows f:a → b from any object a to any object b, a rule for composing arrows f:a → b and g : b → c to obtain an arrow g f : a → c, and for each object A an identity arrow 1_a : a → a. These must satisfy the associative law f(gh) = (fg)h and the left and right unit laws 1_af = f and f1_a = f whenever these composites are defined. In many cases, the objects of a category are best thought of as sets equipped with extra structure, while the morphisms are functions preserving the structure. However, this is neither true for the category of Hilbert spaces nor for the category of cobordisms.

The fundamental idea of category theory is to consider the ‘external’ structure of the arrows between objects instead of the ‘internal’ structure of the objects themselves – that is, the actual elements inside an object – if indeed, an object is a set at all : it need not be, since category theory waives its right to ask questions about what is inside an object, but reserves its right to ask how one object is related to another.

A functor F : C → D from a category C to another category D is a rule which associates to each object a of C an object b of D, and to each arrow f :a → b in C a corresponding arrow F(f): F(a) → F(b) in D. This association must preserve composition and the units, that is, F(fg) = F(f)F(g) and F(1_a) = 1_F(a).

1. Set is the category whose objects are sets, and whose arrows are the functions from one set to another.

2. nCob is the category whose objects are closed (n − 1)-dimensional manifolds Σ, and whose arrows M : Σ₁ → Σ₂ are cobordisms, that is, n-dimensional manifolds having an input boundary Σ₁ and an output boundary Σ₂.

3. Hilb is the category whose objects are Hilbert spaces and whose arrows are the bounded linear operators from one Hilbert space to another.

The ‘new philosophy’ amounts to the following observation: The last two categories, nCob and Hilb, resemble each other far more than they do the first category, Set! If we loosely regard general relativity or geometry to be represented by nCob, and quantum mechanics to be represented by Hilb, then perhaps many of the difficulties in a theory of quantum gravity, and indeed in quantum mechanics itself, arise due to our silly insistence of thinking of these categories as similar to Set, when in fact the one should be viewed in terms of the other. That is, the notion of points and sets, while mathematically acceptable, might be highly unnatural to the subject at hand!

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Austrian Economics. Some Further Ruminations. Part 2.

There are two Austrian theories of capital, at least surfacially with two completely different objectives. The first one concentrates on the physical activities roundabout, time-consuming production processes which are common to all economic systems, and it defines capital as a parameter of production. This theory is considered to be universal and ahistorical, and the present connotation of Austrian Theory of Capital falls congruent with this view. This is often denoted by physical capital and consists of concrete and heterogenous capital goods, which is nothing but an alternative expression for production goods. The second and relatively lesser known theory is the beginning point for a historically specific theory, and shies away with the production process and falls in tune with the economic system called capitalism. Capital isn’t anymore dealing with the production processes, but exclusively with the amount of money invested in a business venture. It is regarded as the central tool of economic calculations by profit-oriented enterprises, and rests on the social role of financial accounting. This historically specific theory of capital is termed business capital and is in a sense simply money invested in business assets.

A deeper analysis, however, projects that these divisions are unnecessary, and that physical capital is not a theory of physical capital at all. Its tacit but implicit research object is always the specific framework of the market economy where production is exercised nearly exclusively by profit-oriented enterprises calculating in monetary terms. Austrian capital theory is used as an element of the Austrian theory of the business cycle. This business cycle theory, if expounded consistently, deals with the way the monetary calculations of enterprises are distorted by changes in the rate of interest, not with the production process as such. In a long and rather unnoticed essay on the theory of capital, Menger (German, 1888) recanted what he had said in his Principles about the role of capital theory in economics. He criticized his fellow economists for creating artificial definitions of capital only because it dovetailed into their personal vision of the task of economics. In respect of the Austrian theory of capital as expounded by himself in his Principles and elaborated on by Böhm-Bawerk, he declared that the division of goods into production goods and consumption goods, important as it may be, cannot serve as a basis for the definition of capital and therefore cannot be used as a foundation of a theory of capital. As for entrepreneurs and lawyers, according to Menger, only sums of money dedicated to the acquisition of income are denoted by this word. Of course, Menger’s real-life oriented notion of capital does not only comprise concrete pieces of money but

all assets of a business, of whichever technical nature they may be, in so far as their monetary value is the object of our economic calculations, i.e., when they calculatorily constitute sums of money for us that are dedicated to the acquisition of income.

An analysis of capital presupposes the historically specific framework of capitalism, characterized by profit-oriented enterprises.

Some economists concluded therefrom that “capital” is a category of all human production, that it is present in every thinkable system of the conduct of production processes—i.e., no less in Robinson Crusoe’s involuntary hermitage than in a socialist society—and that it does not depend upon the practice of monetary calculation. This is, however, a confusion (Mises).

Capital, for Mises, is a device that stems from and belongs to financial accounting of businesses under conditions of capitalism. For him, the term “capital” does not signify anything peculiar to the production process as such. It belongs to the sphere of acquisition, not to the sphere of production.  Accordingly, there is no theory of physical capital as an element or factor in the production process. There is rather a theory of capitalism. For him, the existence of financial accounting on the basis of (business) capital invested in an enterprise is the defining characteristic of this economic system. Capital is “the fundamental notion of economic calculation” which is the foremost mental tool used in the conduct of affairs in the market economy. A more elaborate historically specific theory of capital that expands upon Mises’s thoughts would analyze the function of economic calculation based on business capital in the coordination of plans and the allocation of resources in capitalism. It would not deal with the production process as such but, generally, would concern itself with the allocation and distribution of goods and resources by a system of profit-oriented enterprises.

Conjuncted: Austrian Economics. Some Ruminations. Part 1.

Ludwig von Mises’ argument concerning the impossibility of economic calculation under socialism provides a hint as to what a historical specific theory of capital could look like. He argues that financial accounting based on business capital is an indispensable tool when it comes to the allocation and distribution of resources in the economy. Socialism, which has to do without private ownership of means of production and, therefore, also must sacrifice the concepts of (business) capital and financial accounting, cannot rationally appraise the value of the production factors. Without such an appraisal, production must necessarily result in chaos. 

Austrian Economics. Some Ruminations. Part 1.

Keynes argued that by stimulating spending on outputs, consumption, goods and services, one could increase productive investment to meet that spending, thus adding to the capital stock and increasing employment. Hayek, on the other hand furiously accused Keynes of insufficient attention to the nature of capital in production. For Hayek, capital investment does not simply add to production in a general way, but rather is embodied in concrete capital items. Rather than being an amorphous stock of generalized production power, it is an intricate structure of specific interrelated complementary components. Stimulating spending and investment, then, amounts to stimulating specific sections and components of this intricate structure. Before heading out to Austrian School of Economics, here is another important difference between the two that is cardinal, and had more do with monetary system. Keynes viewed the macro system as vulnerable to periodic declines in demand, and regarded micro adjustments such as wage and price declines as ineffective to restore growth and prosperity. Hayek viewed the market as capable of correcting itself by taking advantages of competitions, and regarded government and Central Banks’ policies to restore growth as sources of more instability.

The best known Austrian capital theorist was Eugen von Böhm-Bawerk, though his teacher Carl Menger is the one who got the ball rolling, providing the central idea that Böhm-Bawerk elaborated. For the Austrians, the general belief lay in the fact that production takes time, and more roundabout the process, the more delay production needs to anticipate. Modern economies comprise complex, specialized processes in which the many steps necessary to produce any product are connected in a sequentially specific network – some things have to be done before others. There is a time structure to the capital structure. This intricate time structure is partially organized, partially spontaneous (organic). Every production process is the result of some multiperiod plan. Entrepreneurs envision the possibility of providing (new, improved, cheaper) products to consumers whose expenditure on them will be more than sufficient to cover the cost of producing them. In pursuit of this vision the entrepreneur plans to assemble the necessary capital items in a synergistic combination. These capital combinations are structurally composed modules that are the ingredients of the industry-wide or economy-wide capital structure. The latter is the result then of the dynamic interaction of multiple entrepreneurial plans in the marketplace; it is what constitutes the market process. Some plans will prove more successful than others, some will have to be modified to some degree, some will fail. What emerges is a structure that is not planned by anyone in its totality but is the result of many individual actions in the pursuit of profit. It is an unplanned structure that has a logic, a coherence, to it. It was not designed, and could not have been designed, by any human mind or committee of minds. Thinking that it is possible to design such a structure or even to micromanage it with macroeconomic policy is a fatal conceit. The division of labor reflected by the capital structure is based on a division of knowledge. Within and across firms specialized tasks are accomplished by those who know best how to accomplish them. Such localized, often unconscious, knowledge could not be communicated to or collected by centralized decision-makers. The market process is responsible not only for discovering who should do what and how, but also how to organize it so that those best able to make decisions are motivated to do so. In other words, incentives and knowledge considerations tend to get balanced spontaneously in a way that could not be planned on a grand scale. The boundaries of firms expand and contract, and new forms of organization evolve. This too is part of the capital structure broadly understood.

Hayek emphasizes that,

the static proposition that an increase in the quantity of capital will bring about a fall in its marginal productivity . . . when taken over into economic dynamics and applied to the quantity of capital goods, may become quite definitely erroneous.

Hayek stresses chains of investments and how earlier investments in the chains can increase the return to the later, complementary investments. However, Hayek is primarily concerned with applying those insights to business cycle phenomena. Also, Hayek never took the additional step that endogenous growth theory has in highlighting the effects of complementarities across intangible investments in the production of ideas and/or knowledge. Indeed, Hayek explicitly excludes their consideration:

It should be quite clear that the technical changes involved, when changes in the time structure of production are contemplated, are not changes due to changes in technical knowledge. . . . It excludes any changes in the technique of production which are made possible by new inventions.

…….

 

Tortile Category and Philosophy of Non-Commutative Geometry

In terms of Feynman diagrams, a quantum field theory is nothing but a finitely generated subcategory QFT of the ribbon category Rep(G). A ribbon category (also called a tortile category) is a braided pivotal category, or equivalently a balanced autonomous category, which satisfies  θ*=θ, where  θ is the twist. This is a kind of category with duals. QFT is generated by the fundamental particles (irreducible representations) and all possible combinations of the fundamental interactions coming from the Feynman rules (intertwiners). Thus one is led to consider generalized quantum field theories QFT′ living inside arbitrary Hermitian ribbon categories R, where the braiding and twist need not be trivial.

Now Tannaka-Krein duality tells us that one can recover the group G from the category Rep(G). In a certain sense, every ribbon category is a category of representations – in the general case not of a group, but of a quantum group. When we do quantum field theory in ribbon categories, we are replacing the symmetry group by a quantum group.

We encounter this phenomenon in Chern-Simons theory, where the Lie group G is replaced by its quantum deformation, Uq(g). The words “Chern–Simons theory” can mean various things to various people, but it generally refers to the three-dimensional topological quantum field theory whose configuration space is the space of  G principal bundles with connection on a bundle and whose Lagrangian is given by the Chern-Simons form of such a connection (for simply connected  G, or rather, more generally, whose action functional is given by the higher holonomy of the Chern-Simons circle 3-bundle. The reason for this deformation of the underlying symmetry group, as one passes from the classical to the quantum theory, has not been altogether elucidated, and remains an interesting problem. In Witten’s approach, three dimensional Chern-Simons theory defines a two dimensional conformal field theory on the boundary, the Wess-Zumino-Witten (WZW) model. The corresponding affine lie algebra g of the WZW model defines, for each k ∈ Z+, a category C_k(g) of integrable modules of level k, and these categories are modular.

On the other hand, in Turaev’s approach, one deforms the lie algebra g into a quantum group U_q(g), where q = e^πi/k, which for k ∈ Z₊ is a root of unity. The representation categories Rep(U_q(g)) of these quantum groups are also modular, and are the starting point in Turaev’s approach.

Despite this theorem, the relationship between the Witten and Turaev approaches is still not completely understood. Ordinary Lie groups are the symmetry groups of manifolds. Quantum groups are the symmetry groups of noncommutative spaces – deformed, noncommutative versions of the commutative algebra of functions on a manifold. Thus the process of passing from QFT to QFT′ is associated with the philosophy of noncommutative geometry, a relatively recent trend in physics.