Closed String Algebra as a Graded-Commutative Algebra C: Cochain Complex Differentials: Part 2, Note Quote.

The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom HY1⊔Y2 ≅ HY1 ⊗ HY2 only makes sense if there is an involutory canonical isomorphism HY1 ⊗ HY2 ≅ HY2 ⊗ HY1 .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V0 ⊕ V1, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)deg v deg ww ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spinc theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spinc structure”.

A spinc structure on a surface with a conformal structure is a pair of holomorphic line bundles L1, L2 with an isomorphism L1 ⊗ L2 ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L1 = L2. On a 1-manifold S a spinc structure means a spinc structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space HS;L1,L2 to each 1-manifold S with spinc structure, and an operator

US0;L1,L2: HS0;L1,L2 → HS1;L1,L2

to each spinc-cobordism from S0 to S1. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin1-manifold (S;L1,L2). Let G = Aut(L1) denote the group of bundle isomorphisms L1 → L1 which cover diffeomorphisms of S. (We can identify this group with Aut(L2).) It has a homomorphism onto the group Diff+(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L1, which can be identified with the group of smooth maps from S to C×. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff+(S) by the commutative Lie algebra Ω0(S) of smooth real-valued functions on S. Let Λ0S;L1,L2 denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ1S;L1,L2 = Γ(L1) ⊕ Γ(L2). The bracket Λ1 ⊗ Λ1 → Λ0 is completely determined by the property that elements of Γ(L1) and of Γ(L2) anticommute among themselves, while the composite

Γ(L1) ⊗ Γ(L2) → Λ0 → VectC(S)

takes (λ12) to λ1λ2 ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L1,L2) to act on the vector space HS;L1,L2, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L1 and L2. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spinc structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spinc-structure.

ε-calculus and Hilbert’s Contentual Number Theory: Proselytizing Intuitionism. Thought of the Day 67.0

Hilbert came to reject Russell’s logicist solution to the consistency problem for arithmetic, mainly for the reason that the axiom of reducibility cannot be accepted as a purely logical axiom. He concluded that the aim of reducing set theory, and with it the usual methods of analysis, to logic, has not been achieved today and maybe cannot be achieved at all. At the same time, Brouwer’s intuitionist mathematics gained currency. In particular, Hilbert’s former student Hermann Weyl converted to intuitionism.

According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.” Whereas the operating with abstract concepts was considered “inadequate and uncertain,” there is a realm of extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.

The objects in questions are signs, both numerals and the signs that make up formulas a formal proofs. The domain of contentual number theory consists in the finitary numerals, i.e., sequences of strokes. These have no meaning, i.e., they do not stand for abstract objects, but they can be operated on (e.g., concatenated) and compared. Knowledge of their properties and relations is intuitive and unmediated by logical inference. Contentual number theory developed this way is secure, according to Hilbert: no contradictions can arise simply because there is no logical structure in the propositions of contentual number theory. The intuitive-contentual operations with signs form the basis of Hilbert’s meta-mathematics. Just as contentual number theory operates with sequences of strokes, so meta-mathematics operates with sequences of symbols (formulas, proofs). Formulas and proofs can be syntactically manipulated, and the properties and relationships of formulas and proofs are similarly based in a logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematical induction or the principle of the excluded middle. These “concept-formations” and modes of reasoning had been criticized by Brouwer and others on grounds that they presuppose infinite totalities as given, or that they involve impredicative definitions. Hilbert’s aim was to justify their use. To this end, he pointed out that they can be formalized in axiomatic systems (such as that of Principia or those developed by Hilbert himself), and mathematical propositions and proofs thus turn into formulas and derivations from axioms according to strictly circumscribed rules of derivation. Mathematics, to Hilbert, “becomes an inventory of provable formulas.” In this way the proofs of mathematics are subject to metamathematical, contentual investigation. The goal of Hilbert is then to give a contentual, meta-mathematical proof that there can be no derivation of a contradiction, i.e., no formal derivation of a formula A and of its negation ¬A.

Hilbert and Bernays developed the ε-calculus as their definitive formalism for axiom systems for arithmetic and analysis, and the so-called ε-substitution method as the preferred approach to giving consistency proofs. Briefly, the ε-calculus is a formalism that includes ε as a term-forming operator. If A(x) is a formula, then εxA(x) is a term, which intuitively stands for a witness for A(x). In a logical formalism containing the ε-operator, the quantifiers can be defined by: ∃x A(x) ≡ A(εxA(x)) and ∀x A(x) ≡ A(εx¬A(x)). The only additional axiom necessary is the so-called “transfinite axiom,” A(t) → A(εxA(x)). Based on this idea, Hilbert and his collaborators developed axiomatizations of number theory and analysis. Consistency proofs for these systems were then given using the ε-substitution method. The idea of this method is, roughly, that the ε-terms εxA(x) occurring in a formal proof are replaced by actual numerals, resulting in a quantifier-free proof. Suppose we had a (suitably normalized) derivation of 0 = 1 that contains only one ε-term εxA(x). Replace all occurrences of εxA(x) by 0. The instances of the transfinite axiom then are all of the form A(t) → A(0). Since no other ε-terms occur in the proof, A(t) and A(0) are basic numerical formulas without quantifiers and, we may assume, also without free variables. So they can be evaluated by finitary calculation. If all such instances turn out to be true numerical formulas, we are done. If not, this must be because A(t) is true for some t, and A(0) is false. Then replace εxA(x) instead by n, where n is the numerical value of the term t. The resulting proof is then seen to be a derivation of 0 = 1 from true, purely numerical formulas using only modus ponens, and this is impossible. Indeed, the procedure works with only slight modifications even in the presence of the induction axiom, which in the ε-calculus takes the form of a least number principle: A(t) → εxA(x) ≤ t, which intuitively requires εxA(x) to be the least witness for A(x).

Derivability from Relational Logic of Charles Sanders Peirce to Essential Laws of Quantum Mechanics

Charles Sanders Peirce made important contributions in logic, where he invented and elaborated novel system of logical syntax and fundamental logical concepts. The starting point is the binary relation SiRSj between the two ‘individual terms’ (subjects) Sj and Si. In a short hand notation we represent this relation by Rij. Relations may be composed: whenever we have relations of the form Rij, Rjl, a third transitive relation Ril emerges following the rule

RijRkl = δjkRil —– (1)

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce considered the individual as the aggregate of all its relations

Si = ∑j Rij —– (2)

The individual Si thus defined is an eigenstate of the Rii relation

RiiSi = Si —– (3)

The relations Rii are idempotent

R2ii = Rii —– (4)

and they span the identity

i Rii = 1 —– (5)

The Peircean logical structure bears resemblance to category theory. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category consists of objects A, B, C,… and arrows (morphisms) f, g, h,… . Each arrow f is assigned an object A as domain and an object B as codomain, indicated by writing f : A → B. If g is an arrow g : B → C with domain B, the codomain of f, then f and g can be “composed” to give an arrow gof : A → C. The composition obeys the associative law ho(gof) = (hog)of. For each object A there is an arrow 1A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, Rij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ Si is Rii.

Rij may receive multiple interpretations: as a transition from the j state to the i state, as a measurement process that rejects all impinging systems except those in the state j and permits only systems in the state i to emerge from the apparatus, as a transformation replacing the j state by the i state. We proceed to a representation of Rij

Rij = |ri⟩⟨rj| —– (6)

where state ⟨ri | is the dual of the state |ri⟩ and they obey the orthonormal condition

⟨ri |rj⟩ = δij —– (7)

It is immediately seen that our representation satisfies the composition rule equation (1). The completeness, equation (5), takes the form

n|ri⟩⟨ri|=1 —– (8)

All relations remain satisfied if we replace the state |ri⟩ by |ξi⟩ where

i⟩ = 1/√N ∑n |ri⟩⟨rn| —– (9)

with N the number of states. Thus we verify Peirce’s suggestion, equation (2), and the state |ri⟩ is derived as the sum of all its interactions with the other states. Rij acts as a projection, transferring from one r state to another r state

Rij |rk⟩ = δjk |ri⟩ —– (10)

We may think also of another property characterizing our states and define a corresponding operator

Qij = |qi⟩⟨qj | —– (11)

with

Qij |qk⟩ = δjk |qi⟩ —– (12)

and

n |qi⟩⟨qi| = 1 —– (13)

Successive measurements of the q-ness and r-ness of the states is provided by the operator

RijQkl = |ri⟩⟨rj |qk⟩⟨ql | = ⟨rj |qk⟩ Sil —– (14)

with

Sil = |ri⟩⟨ql | —– (15)

Considering the matrix elements of an operator A as Anm = ⟨rn |A |rm⟩ we find for the trace

Tr(Sil) = ∑n ⟨rn |Sil |rn⟩ = ⟨ql |ri⟩ —– (16)

From the above relation we deduce

Tr(Rij) = δij —– (17)

Any operator can be expressed as a linear superposition of the Rij

A = ∑i,j AijRij —– (18)

with

Aij =Tr(ARji) —– (19)

The individual states could be redefined

|ri⟩ → ei |ri⟩ —– (20)

|qi⟩ → ei |qi⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨ri |qj⟩ changes and therefore we need an invariant formulation for the transition |ri⟩ → |qj⟩. This is provided by the trace of the closed operation RiiQjjRii

Tr(RiiQjjRii) ≡ p(qj, ri) = |⟨ri |qj⟩|2 —– (22)

The completeness relation, equation (13), guarantees that p(qj, ri) may assume the role of a probability since

j p(qj, ri) = 1 —– (23)

We discover that starting from the relational logic of Peirce we obtain all the essential laws of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement.

Wittgenstein’s Form is the Possibility of Structure

For given two arbitrary objects x and y they can be understood as arguments for a basic ontological connection which, in turn, is either positive or negative. A priori there exist just four cases: the case of positive connection – MP, the case of negative connection – MI, the case that connection is both positive and negative, hence incoherent, denoted – MPI, and the most popular in combinatorial ontology the case of mutual neutrality – N( , ). The first case is taken here to be fundamental.

Explication for σ

Now we can offer the following, rather natural explication for a powerful, nearly omnipotent, synthesizer: y is synthetizable from x iff it is be made possible from x:

σ(x) = {y : MP(x,y)}

Notice that the above explication connects the second approach (operator one) with the third (internal) approach to a general theory of analysis and synthesis.

Quoting one of the most mysterious theses of Wittgenstein’s Tractatus:

(2.033) Form is the possibility of structure.

Ask now what the possibility means? It has been pointed out by Frank Ramsey in his famous review of the Tractatus that it cannot be read as a logical modality (i. e., form cannot be treated as an alternative structure), for this reading would immediately make Tractatus inconsistent.

But, rather ‘Form of x is what makes the structure of y possible’.

Formalization: MP(Form(x), Str(y)), hence – through suitable generalization – MP(x, y).

Wittgensteinian and Leibnizian clues make the nature of MP more clear: form of x is determined by its substance, whereas structurality of y means that y is a complex built up in such and such way. Using syntactical categorization of Lésniewski and Ajdukiewicz we obtain therefore that MP has the category of quantifier: s/n, s – which, as is easy to see, is of higher order and deeply modal.

Therefore M P is a modal quantifier, characterized after Wittgenstein’s clue by

MP(x, y) ↔ MP(S(x), y)