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 S_iRS_j between the two ‘individual terms’ (subjects) S_j and S_i. In a short hand notation we represent this relation by R_ij. Relations may be composed: whenever we have relations of the form R_ij, R_jl, a third transitive relation R_il emerges following the rule

R_ijR_kl = δ_jkR_il —– (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

S_i = ∑_j R_ij —– (2)

The individual S_i thus defined is an eigenstate of the R_ii relation

R_iiS_i = S_i —– (3)

The relations R_ii are idempotent

R²_ii = R_ii —– (4)

and they span the identity

∑_i R_ii = 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 1_A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, R_ij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ S_i is R_ii.

R_ij 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 R_ij

R_ij = |r_i⟩⟨r_j| —– (6)

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

⟨r_i |r_j⟩ = δ_ij —– (7)

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

∑_n|r_i⟩⟨r_i|=1 —– (8)

All relations remain satisfied if we replace the state |r_i⟩ by |ξ_i⟩ where

|ξ_i⟩ = 1/√N ∑_n |r_i⟩⟨r_n| —– (9)

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

R_ij |r_k⟩ = δ_jk |r_i⟩ —– (10)

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

Q_ij = |q_i⟩⟨q_j | —– (11)

with

Q_ij |q_k⟩ = δ_jk |q_i⟩ —– (12)

and

∑_n |q_i⟩⟨q_i| = 1 —– (13)

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

R_ijQ_kl = |r_i⟩⟨r_j |q_k⟩⟨q_l | = ⟨r_j |q_k⟩ S_il —– (14)

with

S_il = |r_i⟩⟨q_l | —– (15)

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

Tr(S_il) = ∑_n ⟨r_n |S_il |r_n⟩ = ⟨q_l |r_i⟩ —– (16)

From the above relation we deduce

Tr(R_ij) = δ_ij —– (17)

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

A = ∑_i,j A_ijR_ij —– (18)

with

A_ij =Tr(AR_ji) —– (19)

The individual states could be redefined

|r_i⟩ → e^iφ_i |r_i⟩ —– (20)

|q_i⟩ → e^iθ_i |q_i⟩ —– (21)

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

Tr(R_iiQ_jjR_ii) ≡ p(q_j, r_i) = |⟨r_i |q_j⟩|² —– (22)

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

∑_j p(q_j, r_i) = 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.

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