Quantum machanics and Quantum logic

Haruhiko Kozawa

                         matrix@isc.chubu.ac.jp

Graduate School of Engineering, Chubu University

Introduction

We define a propositional system realizing a physical system. The propositional system is an orthomodular lattice P (H)of projections on Hilbert space H, whose non-distributivity is explained by the experiment of electron spin. We consider reals and propositions on the P (H)-valued universe V^{P (H)}.

1 Propositional system and Hilbert space

In the physical system, the set L of propositons satisfys the following axoims (C), (P), (A) .

(C) For each proposition c ∈ L there exists a unique orthocomplement c^{⊥} ∈ L such that (C_{1}) c^{⊥⊥} = c (C_{2}) c ∨ c^{⊥} = 1, and c ∧ c^{⊥} = 0 (C_{3}) b < c ⟹ c^{⊥} < b^{⊥} for ∀b ∈ L

(P) If b < c are propositions of L, then the sublattice generated by {b, b^{⊥}, c, c^{⊥}} is a classical propositional system, that is to say, a distributive lattice.

(A) If b ≠ c and b < c, one say that c covers b when

b < x < c ⟹ x = b or x = c.

An element which covers 0 is called an atom.

(A_{1}) If b is a proposition different from 0, there exists an atom p < b.

(A_{2}) If p is an atom and if p ∧ b = 0, then p ∨ b covers b.

Then, we define follwing difinition.

Definition 1 A complete lattice satisfying axioms (C),(P),(A) is called a propositional sytem.

Definition 2 In a propositional system L, two propositions b and c are said to be compatible if the sublattice generated by \{b, b^{⊥}, c, c^{⊥}\} is distributive, and denoted by b\begin{matrix}| \\ ∘ \end{matrix} c.

Let P (H) be the set of projections of Hilbert space H, which is a propositional system . If P ∈ P (H), then the eigenvalues of P is 1 or 0. Eigenvalue 1 corresponds to true and 0 to false. If P_{1}, P_{2} ∈ P (H) and K = {x | P_{1}x = x}, L = {x | P_{2}x = x}, we define

1. P_{1} < P_{2} if and only if K ⊂ L

2. P_{1} ∨ P_{2} is the projection of H onto K ∨ L. (K ∨ L is a subspace spanned by K ∪ L.)

3. P_{1} ∧ P_{2} is the projection of H onto K ∩ L.

4. P_{1}^{⊥} is the projection of H onto K^{⊥}. (P_{1}^{⊥} = (I − P_{1}))

Then, P (H) is a complete lattice satisfying axioms (C),(P),(A). In P (H), commutativity is completely equivalent to compatibility.

2 Quantum Logic for electron spin

Now, we consider the electron spin in the logic of P (H). When we measure deflection produced by a strong magnetic feild in a direction orthogonal to the direction of the path of the electron, we have exactly two outcomes. There is a variable called the “spin” variable that is associated with how the electron is deflected by magnets. (We call two outcomes spin-up and spin-down, respectively.) We know that the components of spin can’t be determined in two direction simultaneously. For this reason, we must abandon the distributivity axiom which we postulated for classical propositional system.

Next, we shall state the Hilbert space corresponding to the propositional system. Let x-y plane be the plane which is orthogonal to the path of the electron and x axis is the positive horizontal axis, y axis is the vertical axis. For each axis w, let θw be the angle measured counterclockwise between the x axis and the w axis. Let Kw be a subspace which is corresponding to “w-spin-up” and Kw^{⊥} to “w-spin-down”.

The 2 dimensional Hilbert space H provides an description of propositional syatem for electron spin as follows; Kw is the one-dimensional subspace of H oriented at angle θ_{w}/2 from the positive horizontal axis and Kw^{⊥} is the orthogonal complement of Kw. The spin state of an electron can be represented by a unit vector ψ in the Hilbert space H. The probability ω of obtaining the outcome “w-spin-up” is the square of the length of the projection of ψ onto the subspace K_{w}, and the probability ω of obtaining the outcome “w-spin-down” is the square of the length of the projection of ψ onto the subspace K_{w}^{⊥}.

ω(w-spin-up) = ⟨P_{K_{w}}ψ, ψ⟩, ω(w-spin-down) = ⟨P_{K_{w}^{⊥}}ψ, ψ⟩

Then, we have the set of projections on H,

P (H) = {I, 0, P_{Kw}, P_{Kw}^{⊥} | 0 ≤ θw < π}.

3 On P (H)-valued universe V^{P (H)}

Let P (H) be a propositional system. P (H)-valued universe V^{P (H)} is constructed by induction.

V_{α}^{P (H)} = {u | ∃β <α ∃Du⊂V_{β}^{P (H)}(u : Du → P (H))}, V^{P (H)} = ⋃_{α∈On} V_{α}^{P (H)}.

Implication →_{q} is defined by

φ →_{q} ψ \underleftrightarrow{def} φ^{⊥} ∨ (φ ∧ ψ)

For u, v ∈ V^{P (H)}, [[u =_{q} v]] and [[u ∈_{q} v]] are defined by

[[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),

where [[A]] is an element of P (H).

Next, we turn to real numbers in V^{P (H)}.

Definition 3 u is called a real number if (1)∧(2)∧(3), where

1. u ⊂ ℚ̌, that is, ∀x(x ∈ u →_{q} x ∈ ℚ̌)

2. ∃r ∈ ℚ̌(r ∈ u) ∧ ∃r ∈ ℚ̌(r ∈ u)^{⊥}

3. ∀r ∈ ℚ̌((r ∈ u) ↔_{q} ∀ ∈ ℚ̌(r < s →_{q} s ∈ u))

Theorem 4 (cf. [3]) If u is a real in V^{P (H)}, then E_{u} : ℝ → P (H) defined by E_{u}(λ) = ⋀_{λ<r}[[ř ∈ u]] is a resolution of the identity, i.e. E_{u} satisfies ⋀_{λ∈ℝ} E_{u}(λ) = 0 and ⋁_{λ∈ℝ} E_{u}(λ) = 1, E_{u}(λ) = ⋀_{λ<μ} E_{u}(μ), and gives the spectral representation of the corresponding self-adjoint operator A = ∫ λdE_{u}(λ) on the Hilbert space H.

Conversely, if A is a self-adjoint operator A = ∫ λdE(λ)on H, we define u by Du = {ř | r ∈ ℚ}, u(ř) = ⋀_{r<s} E(s). Then u is a real in V^{P (H)}, and A = ∫ λdE_{u}(λ).

4 P (H)-valued universe V^{P (H)} for electron spin

We consider x-spin and y-spin, that is, propositional system P (H) = {I, 0, P_{Kx}, P_{Kx}^{⊥}, P_{Ky}, P_{Ky}^{⊥}}.

This lattice includes Boolean subalgebras B_{1}, B_{2}, where B_{1} = {I, 0, P_{Kx}, P_{Kx}^{⊥}}, B_{2} = {I, 0, P_{Ky}, P_{Ky}^{⊥}}.

Thus, V^{P (H)} includes V^{B_{1}} and V^{B_{2}}.

Unless otherwise noted, H is the same Hilbert space of dimension 2 as in section 2. Hence, if A is a self-adjoint operator on H, then

A = λ_{1}P + λ_{2}P^{⊥}.

4.1 real numbers

Let u be a real in V^{P (H)} and A_{u} a self-adjont operator corresponding to u. For λ̌, we have

Aλ̌ = λI.

If [[u is real]] = I, [[u =_{q} λ̌_{1}]] = P_{Kx} and [[u =_{q} λ̌_{2}]] = P_{Kx}^{⊥} , we have

A_{u} = λ_{1}P_{Kx} + λ_{2}P_{Kx}^{⊥}.

Especially, if A_{u} is a proposition “x-spin-up”in P (H), that is, A_{u} = P_{Kx}, then,

[[u =_{q} ˇ1]] = [[u =_{q} ˇ0]]^{⊥} = P_{Kx}.

Then, u is a real satisfying (u =_{q} ˇ1) ↔ (u =_{q} ˇ0)^{⊥} in V^{P (H)} . ˇ1(š) is 0 or 1, and hence u(ř) is compatible

with ˇ1(š), ˇ0(š).

Thus, we define the following definition.

Definition 5 Proposition u in V^{P (H)} is a real number satisfying (u =_{q} ˇ1) ↔ (u =_{q} ˇ0)^{⊥}.

References

[1] C.Piron, Fundations of Quantum Pysics, W.A.Benjamin, Inc. 1976

[2] David W. Cohen, An Introduction to Hilbert space and Quantum Logic, Springer-Verlag New York Inc. 1989

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

[4] G.Takeuti, Quantum Set Theory, Current Issues in Quantum Logic, eds. E.Beltrametti and B.C.van Frassen, Plenum,New York (1981) pp.303-322

[5] 竹内外史線形代数と量子力学裳華房 1981

[6] S.Titani, Lattice Valued Set Theory, Arch. Math. Logic 38-6(1999) pp.395-421

[7] R.I.G. ヒューズ量子論理