Quantum Set Theory

Satoko Titani

                           titani@chubu.ac.jp

Chubu University

1 Introduction

Quantum physics is interpreted in a Hilbert space model : Physical states are represented as vectors in a Hilbert space, and observables are represented as self-adjoint operators on the Hilbert space. Let H be a Hilbert space realizing a physical system and let Q(H) be the complete orthomodular lattice consisting of closed linear subspaces of H, where

⋁_{α} p_{α} = \overline{⋃_{α} p_{α}},
⋀_{α} p_{α} = ⋂_{α} p_{α},
p^{⊥} = the orthocomplement of p.

Q(H) is the algebraic expression of quantum logic on the physical system.

We deal with a theory on Q(H)-valued sets, in classical set theory. Let V be the universe of classical set theory as our standpoint, and Q = Q(H). Q-valued sets and Q-valued universe V^{Q} consisting of all Q-valued sets are constructed in V. We call V the external universe for V^{Q}. Set theory on the Q-valued universe V^{Q} is called quantum set theory.

In [1] and [2], G.Takeuti developed the quantum set theory and showed that each real number in V^{Q}, which is defined as a Dedekind cut of rational numbers, represents a self-adjoint operator in the external universe V. We call these reals in V^{Q} q-reals. Thus, observables are represented by q-reals. G.Takeuti defined addition + and multiplication · for commutative q-reals and showed that they represent the corresponding operations on the set of self-adjoint operators. Even if self-adjoint operators u, v are not commutative, u + v and u · v are defined as linear operators.

We develop Takeuti’s quantum set theory by adding a strong implication to the logic, and extend the concept of q-reals and q-complexes.

2 Quantum Logic

Let C be a property of the states of a physical system S. C is considered as an observable as follows: Each measurement which distinguished between the presence or absence of C is considered as a measurement of this observable, such that its value is 1 if C is verified, and zero in the opposite case. The observable that is identified with property C is represented by a projection [[C]] on H. Namely, [[C]] is the projection on the closed subspace consisting of all states φ which possess the property C with certainty. The closed subspace which is the range of [[C]] will also denoted by [[C]] without any confusion. [[C]] is an element of Q, and is also a q-real in V^{Q} as a projection.

Now we consider the property C as a proposition with the truth value [[C]], and logical operators ∧, ∨ are interpreted as lattice operators ∧, ∨ on Q. Complete orthomodular lattice Q = Q(H) consisting of closed subspaces of a complex Hilbert space H is called quantum logic. Q is irreducible, that is, the center Z(Q) of Q consists of two elements 0 and 1. We define implication → on Q by

(p → q) {def \atop =} ⋁\{x∈Z(Q) | p ∧ x ≤ q\} = \{ \begin{matrix}1 & if p ≤ q \\ 0 & otherwise. \end{matrix}

This implication represents the lattice order ≤ on Q, and is called the basic implication.

Generally, lattice valued universe V^{L} is constructed for any complete lattice L. In [3], we showed that a lattice valued set theory on the lattice valued universe V^{L} can be developed by introducing the basic implication →.

Our quantum logic has, as primitive logical symbols, ∧, ∨, ∀, ∃, orthogonal complementation ^{⊥}, and basic implication →.

3 Quantum set theory

Quantum valued universe V^{Q}, consisting of all Q-valued sets, is constructed in the same way as Boolean valued universe V^{B} in the external universe V of classical set theory. G.Takeuti developed a quantum set theory in [1] and [2] using the notation p → q by which he denotes p^{⊥} ∨ (p ∧ q). We call p^{⊥} ∨ (p ∧ q) quantum implication, and denote it by →_{Q} in order to distinguish from the basic implication →. That is,

p →_{Q} q \underleftrightarrow{def} p^{⊥} ∨ (p ∧ q).

→_{Q} is weaker than → in the sense that (p → q) ≤ (p →_{Q} q). Equality and membership relation =_{Q} and ∈_{Q} corresponding to the quantum implication are interpreted on the universe V^{Q} in [1] and [2] as follows :

[[u =_{Q} v]] = ⋀_{x∈Du}(u(x) →_{Q} [[x ∈_{Q} v]]) ∧ ⋀_{x∈Dv}(v(x) →_{Q} [[x ∈_{Q} u]])
[[u ∈_{Q} v]] = ⋁_{x∈Dv}[[u =_{Q} x]] ∧ v(x)

For the quantum equality =_{Q}, the axiom of equality is not valid, because of non-distributivity of Q. Distributive law

p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)

holds only in the case that p commutative with both of q and r, where “p commutative with q” means that

p = (p ∧ q) ∨ (p ∧ q^{⊥}).

Thus, the axiom of equality for =_{Q} is restricted. This means that the quantum equality =_{Q} is not strong enough to be qualified for an equality.

So, as primitive predicate symbles, we introduce = and ∈ corresponding to the basic implication →. They are interpreted on V^{Q} as

[[u=v]] = ⋀_{x∈Du}(u(x) → [[x∈v]]) ∧ ⋀_{x∈Dv}(v(x) → [[x∈u]])
[[u∈v]] = ⋁_{x∈Dv}[[u=x]] ∧ v(x).

These = and ∈ are called the basic equality and the basic membership relation. The quantum equality =_{Q} and the quantum membership relation ∈_{Q} will be defined in our quantum set theory.

We use the following abbreviations.

φ →_{Q} ψ \underleftrightarrow{def} φ^{⊥} ∨ (ψ ∧ φ)
φ\begin{matrix}| \\ ∘ \end{matrix}ψ \underleftrightarrow{def} φ → (φ ∧ ψ) ∨ (φ ∧ ψ^{⊥})
\begin{matrix}| \\ ∘ \end{matrix}(Γ) \underleftrightarrow{def} ⋀_{φ,ψ∈Γ} φ\begin{matrix}| \\ ∘ \end{matrix}ψ

φ \underleftrightarrow{def} (φ → φ) → φ

Then the following nonlogical axioms GA1–GA11 of lattice valued set theory LZFZ in [3] are valid on V^{Q}.

GA1. Equality ∀u∀v (u = v ∧ φ(u) → φ(v)) .

GA2. Extensionality ∀u, v (∀x(x∈u ↔ x∈v) → u = v).

GA3. Pairing ∀u, v∃z (∀x(x∈z ↔ (x = u ∨ x = v))).

The set z satisfying ∀x(x∈z ↔ (x = u ∨ x = v)) is denoted by {u, v}.

GA4. Union ∀u∃z (∀x(x∈z ↔ ∃y ∈u(x∈y))) .

The set z satisfying ∀x(x∈z ↔ ∃y ∈u(x∈y)) is denoted by ⋃ u.

GA5. Power set ∀u∃z (∀x(x∈z ↔ x ⊂ u)), where

x⊂u \underleftrightarrow{def} ∀y(y ∈x → y ∈u).

The set z satisfying ∀x(x∈z ↔ x⊂u) is denoted by P(u).

GA6. Infinity ∃u (∃x(x∈u) ∧ ∀x(x∈u → ∃y ∈u(x∈y))) .

GA7. Separation ∀u∃v (∀x(x∈v ↔ x∈u ∧ φ(x))).

The set v satisfying ∀x(x∈v ↔ x∈u ∧ φ(x)) is denoted by {x∈u | φ(x)}.

GA8. Collection

∀u∃v (∀x(x∈u → ∃yφ(x, y)) → ∀x(x∈u → ∃y ∈vφ(x, y))).

GA9. ∈-induction ∀x (∀y(y ∈ x → φ(y)) → φ(x)) → ∀xφ(x).

GA10. Zorn Gl(u) ∧ ∀v (Chain(v, u) → ⋃ v ∈ u) → ∃zMax(z, u), where

Gl(u) \underleftrightarrow{def} ∀x(x ∈ u → x ∈ u),
Chain(v, u) \underleftrightarrow{def} v ⊂u ∧ ∀x, y(x, y ∈v → x⊂y ∨ y ⊂x),
Max(z, u) \underleftrightarrow{def} z ∈ u ∧ ∀x(x∈u ∧ z ⊂x → z = x).

GA11. Axiom of ♢ ∀u∃z∀t(t∈z ↔ ♢(t∈u)).

The set z satisfying ∀t(t∈z ↔ ♢(t∈u)) is denoted by ♢u.

We take GA1–GA11 as nonlogical axioms of quantum set theory. Then the induction principle is provable in the quantum set theory, and enables us to define the quantum equality =_{Q} and quantum membership relation ∈_{Q}, inductively.

4 Quantum numbers

The external universe V, in which V^{Q} is constructed, can be embedded in V^{Q} :

u ↦ ǔ ∈ V^{Q} for u∈V

The set of rational numbers in V^{Q} is Q̌ corresponding to the set of rational numbers Q in V by the embedding. Real numbers are defined as Dedekind cuts of Q̌. (cf.[1]). Since V^{Q} has more members than V̌, Dedekind cuts of Q̌ in V^{Q} are more rich than those in V. The set of q-reals is denoted by R^{Q}. Q-reals represent self-adjoint operators on the Hilbert space H, hence q-reals represent observables in the physical system. If u, v are commutative q-reals, then u + v and u · v are defined in R^{Q}, as shown in [1]. Q-complexes are defined as u + iv where u, v are commutative q-reals. Each q-complex represents normal operator on H. The set of all q-complexes is denoted by C^{Q}, and the set of all bounded q-complexes is denoted by C_{b}^{Q}.

Let L(H) be the set of closed and densely defined linear operators on H. Self-adjoint operators and normal operators are in L(H), that is, R^{Q} and C^{Q} are subsets of L(H). We extend the concept of numbers such as q-reals and q-complexes to closed and densely defined linear operators. For that, we represent H as a Hilbert space H in V^{Q}.

For a q-real A, that is, a self-adjoint operator A on H, there exists a unique resolution E of the identity such that A = ∫ λdE(λ). Let E be the set of all resolutions of the identity. We can show that for each resolution of the identity, E ∈E, there exists an isomorphism of H into C^{Q} :

σ_{E}:H→C^{Q}

σ_{E}(H) is a Hilbert space over Č. It follows that each vector x in H is considered as a mapping of E into C^{Q} :

x̂:E →C^{Q} by x̂(E) = σ_{E}(x)

For each E ∈E, let

H(E) = {x̂(E) | x∈H}
H_{b}(E) = H(E) ∩ C_{b}^{Q}

H(E) is isomorphic to H as Hilbert space. H_{b}(E) is dense in H(E) and forms an abelian von Neumann algebra. Each u∈H(E) defines a multiplication operator û on H_{b}(E) :

û : x̂(E) ↦ (ux)ˆ (E) = u · x̂(E)

We represent H as the set of x̂∈Π_{E∈E}H(E)∈V^{Q}, and denote it by H.

H = {x̂:E →C^{Q} | x∈H}⊂Π_{E∈E}H(E)∈V^{Q}.

Thus, q-real (or q-complex) ∫ λdE(λ) is in H(E), and represents a self-adjoint operator (or normal operator) on H.

H can be defined in the language of quantum set theory, and q-reals (or q-complexes) are considered as closed and densely defined linear operators on H. Let L(H) be the set of all closed and densely defined linear operators on H. Members of L(H) we call q-numbers.

References

[1] G.Takeuti, Two Applications of Logic to Mathematics, Iwanami and Princeton University Press, Tokyo and Princeton (1978).

[2] G.Takeuti, Quantum Set Theory, Current Issues in Quantum Logic, eds. E.Beltrametti and B.C.van Frassen, Plenum,New York (1981) p.303

[3] S.Titani, Lattice Valued Set Theory, Arch. Math. Logic vol 38-6(1999).