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Measurement theory is, by an analogy of quantum mechanics (or, as a linguistic turn of quantum mechanics ), constructed as the scientific theory formulated in a certain C -algebra (i.e., a norm ∗ A closed subalgebra in the operator algebra B(H) composed of all bounded linear operators on a Hilbert space H, cf. [24, 26] ). Let be the weak closure of , which is called a W -algebra. N ∗ A ∗ The structure [ B(H)] is called a fundamental structure of MT. A⊆N ⊆ When = (H), the C -algebra composed of all compact operators on a Hilbert space H, the A C ∗ MT is called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when is commutative that is, when A A is characterized by C0(Ω), the C∗-algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space Ω (cf. [26]) , the MT is called classical measurement theory. Thus, we have the following classification: quantum MT (when non-commutative = (H), = B(H)) A C N (A) MT classical MT (when commutative = C (Ω)), = L (Ω,ν)( B(L2(Ω,ν))) A 0 N ∞ ⊆
Also, we assert that quantum language is located as follows:
realistic view relativity (unsolved) theory 3 theory of (monism) → −−−−−−→ 5 Newton 2 everything Parmenides −−→ −−→ (realism) (quantum phys.) Socrates quantum mechanics 4 Greek Schola- → −−−−−−→ philosophy 1 −−−−−→sticism language Plato (dualism) 7 −−−−−−→ (=MT) Alistotle (linguistic view) Descartes quantum Rock,... language 10 6 quanti- language Kant philosophy 8 −−→ −−→ (idealism) −−→ −−−−−→zation (language) statistics system theory 9 −−−−→
linguistic view Figure 1: The history of the world-view
1.2 Observables Let [ B(H)] be the fundamental structure of measurement theory. Let be the A⊆N ⊆ N∗ pre-dual Banach space of . That is, = ρ ρ is a weak∗ continuous linear functional on N N∗ { | N , and the norm ρ is defined by sup ρ(F ) : F such that F (= F B(H)) 1 . The } k kN∗ {| | ∈N k kN k k ≤ } bi-linear functional ρ(F ) is also denoted by ρ, F , or in short ρ, F . Define the mixed state N∗ h iN h i ρ ( ) such that ρ = 1 and ρ(F ) 0 for all F satisfying F 0. And put ∈N∗ k kN∗ ≥ ∈N ≥ Sm( )= ρ ρ is a mixed state . N∗ { ∈N∗ | } 3
According to the noted idea (cf. ref. [3]) in quantum mechanics, an observable O (X, , F ) ≡ F in the W -algebra is defined as follows: ∗ N (B ) [σ-field] X is a set, ( 2X , the power set of X) is a σ-field of X, that is, “Ξ , Ξ , Ξ , 1 F ⊆ 1 2 3 ··· ∈ Ξ ”, “X ” and “Ξ X Ξ ”. F⇒∪k∞=1 k ∈ F ∈ F ∈F⇒ \ ∈ F (B ) [Countably additivity] F is a mapping from to satisfying: (a): for every Ξ , F (Ξ) 2 F N ∈ F is a non-negative element in such that 0 F (Ξ) I, (b): F ( ) = 0, where 0 and I N ≤ ≤ ∅ is the 0-element and the identity in respectively. (c): for any countable decomposition N Ξ , Ξ ,... of Ξ (i.e., Ξ , Ξ such that ∞ Ξ =Ξ, Ξ Ξ = (i = j)), it holds { 1 2 } ∈ F k ∈ F k=1 k i ∩ j ∅ 6 that S
K m lim ρ, F ( Ξk) = ρ, F (Ξ) ( ρ S ( )) (2) K N∗ h iN N∗ h iN ∀ ∈ N∗ →∞ k[=1
K i.e., limK F ( k=1 Ξk)= F (Ξ) in the sense of weak∗ convergence in . →∞ N S Remark 1. In the above (b), it is usual to assume the condition: F (X) = I. In fact, through all this paper except Section 5, the condition: F (X) = I is assumed. However, for the reason mentioned in Remark 9 later, we start from the above (b).
1.3 Quantum language ( Axioms ) With any system S, a fundamental structure [ B(H)] can be associated in which A⊆N ⊆ the pure measurement theory (A1) of that system can be formulated. A pure state of the system S is represented by an element ρp( Sp( )=”pure state class”(cf. ref. [14])) and an observ- ∈ A∗ able is represented by an observable O =(X, , F ) in . Also, the measurement of the observ- O Fp N M O able for the system S with the pure state ρ is denoted by ( ,S[ρp]) or more precisely, M O N ( =(X, , F ),S[ρp]) . An observer can obtain a measured value x ( X) by the measurement MN O F ∈ ( ,S[ρp]). N The Axiom 1 presented below is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics. Axiom 1 [Pure Measurement]. The probability that a measured value x ( X) obtained by the ∈ p M O X, , F , S p ρ F F measurement ( ( ) [ρ0]) belongs to a set Ξ( ) is given by 0( (Ξ)), if (Ξ) is N ≡ pF ∈ F essentially continuous at ρ0 (cf. ref. [14]). Next, we explain Axiom 2 in (A ). Let (T, ) be a tree, i.e., a partial ordered set such that 1 ≤ “t1 t3 and t2 t3” implies “t1 t2 or t2 t1”. Assume that there exists an element t0 T , ≤ ≤ ≤ ≤ 2 2 ∈ called the root of T , such that t0 t ( t T ) holds. Put T = (t1,t2) T t1 t2 . The ≤ ∀ ∈ ≤ { ∈ | ≤ } 2 family Φt1,t2 : t2 t1 (t1,t2) T is called a causal relation (due to the Heisenberg picture), if it { N →N } ∈ ≤ satisfies the following conditions (C1) and (C2). (C ) With each t T , a fundamental structure [ B(H )] is associated. 1 ∈ At ⊆Nt ⊆ t 2 t1,t2 (C2) For every (t1,t2) T , a continuous Markov operator Φ : t2 (with the weak∗ topology) ∈ ≤ t1,t2 Nt1,t2 t1 (with the weak∗ topology) is defined (i.e., Φ 0, Φ (I t2 ) = I t1 ). And it → N t1,t2 t2,t3 t1,t3 ≥ 2 N N satisfies that Φ Φ =Φ holds for any (t1,t2), (t2,t3) T . ∈ ≤ 4
t1,t2 Sm Sm 2 The family of pre-dual operators Φ : (( t1 ) ) (( t2 ) ) (t1,t2) T is called a pre- { ∗ N ∗ → N ∗ } ∈ t≤1,t2 p dual causal relation (due to the Schr¨odinger picture). If we can regard that Φ (S (( t1 )∗)) ∗ A ⊆ Sp 2 (( t2 )∗), the causal relation Φt1,t2 : t2 t1 (t1,t2) T is said to be deterministic. A { N →N } ∈ Now Axiom 2 in the measurement theory (1) is presented≤ as follows: 1 2 t ,t 2 Axiom 2 [Causality]. The causality is represented by a causal relation Φ : t2 t1 (t1,t2) T . { N →N } ∈ ≤
1.4 Linguistic Interpretation Next, we have to study the linguistic interpretation (i.e., the manual of how to use the above axioms, ) as follows. That is, we present the following interpretation (D) [=(D1), (D2)], which is characterized as a kind of linguistic turn of so-called Copenhagen interpretation (cf. refs. [14,15] ). That is,
(D1) Consider the dualism composed of “observer” and “system( =measuring object)”. And there- fore, “observer” and “system” must be absolutely separated.
(D2) Only one measurement is permitted. And thus, the state after a measurement is meaningless since it can not be measured any longer. Therefore, the wave collapse is prohibited. Also, the causality should be assumed only in the side of system, however, a state never moves. Thus, the Heisenberg picture should be adopted. And thus, the Schr¨odinger picture is rather makeshift. Thus, the problem ”when and where a measurement is performed?” is nonsense. and so on. For example, the axioms seem the rule of how to move the piece of a chess game. On the other hand, the linguistic interpretation resembles the standard tactics of chess game. In this sense, we cannot completely say all about the linguistic interpretation. The following argument is a consequence of the above (D2). For each k = 1, 2,...,K, consider a measurement M (Ok (Xk, k, Fk),S[ρ]). However, since the (D2) says that only one measurement N ≡ F M O K is permitted, the measurements ( k,S[ρ]) should be reconsidered in what follows. Under { N }k=1 the commutativity condition such that
Fi(Ξi)Fj(Ξj)= Fj(Ξj)Fi(Ξi) (3) ( Ξ , Ξ , i = j), ∀ i ∈ Fi ∀ j ∈ Fj 6 we can define the product observable (or, simultaneous observable) K O = ( K X , ⊠K , k=1 k k=1 k k=1Fk K F ) in such that × × ×k=1 k N ( K F )( K Ξ )= F (Ξ )F (Ξ ) F (Ξ ) ×k=1 k ×k=1 k 1 1 2 2 · · · K K ( Ξ , k = 1,...,K). ∀ k ∈ Fk ∀ ⊠K K Here, k=1 k is the smallest σ-field including the family k=1Ξk :Ξk k k = 1, 2,...,K . F M O K {× ∈ F } Then, the above ( k,S[ρ]) k=1 is, under the commutativity condition (3), represented by the { N M } K O simultaneous measurement ( k=1 k, S[ρ]). N × Consider a finite tree (T t0,t1,...,tn , ) with the root t0. This is also characterized by ≡{ } ≤ t,t the map π : T t T such that π(t) = max s T s O O t,t′ 2 The pair [ T ] = [ t t T , Φ : t′ t (t,t ) T ] is called a sequential causal observable. { } ∈ { N → N } ′ ∈ ≤ O ⊠ For each s T , put Ts = t T t s . And define the observable s ( t Ts Xt, t Ts t, Fs) ∈ { ∈ | ≥ } ≡ ∈ ∈ F in as follows: × Ns b b Os (if s T π(T ) ) Os = ∈ \ (4) O ( 1 Φ O ) (if s π(T )) s t π− ( s ) π(t),t t b × × ∈ { } ∈ b O O if the commutativity condition holds (i.e., if the product observable s ( t π 1( s ) Φπ(t),t t) × ∈ − { } exists) for each s π(T ). Using (4) iteratively, we can finally obtain the observabl× e O in . ∈ t0 Nbt0 The O is called the realization (or, realized causal observable) of [O ]. t0 T b b Remark 2 [Particle or wave]. The argument about the ”particle vs. wave” is meaningless in quantum language. As seen in the following table, this argument is traditional: Theories P or W Particle(=symbol) Wave(= mathematical representation ) \ Aristotles hyle eidos Newton mechanics point mass state (=(position, momentum)) Statistics population parameter Quantum mechanics particle state ( wave function) ≈ Quantum language system (=measuring object) state In the above table, Newtonian mechanics (i.e., mass point state) may be easiest to understand. ↔ Thus, ”particle” and ”wave” are not confrontation concepts. In this sense, the ”wave or particle” is meaningless. In the linguistic interpretation of quantum mechanics, this should be usually understood as the problem ”interference or no interference”. Remark 3 [Reality]. Since quantum language is a kind of metaphysics, we are not concerned with the reality such as discussed in [4] and [2]. Also, since space and time are independent in quantum language (cf. [15] ), we can not expect it to yield a good physical theory (i.e., 5 in Figure 1). Remark 4 [The Schr¨odinger’s cat]. Axiom 2 allows us to deal with more than the deterministic causal relation, for example, the Brownian motion and the quantum decoherence, etc. Therefore, we can easily describe the Schr¨odinger’s cat by quantum language. Thus, this is not a paradox in quantum language. However, quantum language (due to dualism composed of “observer” and “system”) does not have a power to describe Wigner’s friend as well as Descartes’ proposition ”I think, therefore I am” (cf. [16]). Remark 5 [The commutative condition (3)].(i): If the commutative condition (3) does not hold, K O = ( K X , ⊠K , K F ) is not an observable, but the -valued measure space. ×k=1 k ×k=1 k k=1Fk ×k=1 k N (ii): Assume that there exists an observable O = ( K X , ⊠K , F ) such that ×k=1 k k=1Fk b b F (X1 X2 Xj 1 Ξj Xj+1 XK)= Fj(Ξj) (Ξj j, j = 1, 2, ..., K) × ×···× − × × ×··· ∈ F Then, thereb is a reason to call the O a simultaneous observable. (iii): Also, it may be worth while investigating the concept such that O = ( K X , ⊠K , F ) b k=1 k k=1Fk is an simultaneous observable concerning ρ. × b b 2 The double-slit experiment Although Feynmann’s enthusiasm is transmitted in the explanation of the double-slit experiment in [6], we do not think that his explanation is sufficient. That is because the double-slit experiment and so on should be explained after the answer to ”What kind of measurement is taken?”. 6 y ✻ A (b, 2δ) (b, δ) P ✲ x a b →• (b, δ) − B Figure 2(1). Potential V (x,y)= on the thick line, = 0 (elsewhere) 1 ∞ y ✻ A (b, 2δ) (b, δ) P ✲ x a b →• (b, δ) − B Figure 2(2). Potential V (x,y)= on the thick line, = 0 (elsewhere) 2 ∞ That is, V2 = V1 + ”the line segment ab” Consider a tree (T, ) with the two branches such that ≤ T = 0 T T { }∪ 1 ∪ 2 where T = (1,s) s> 0 , T = (2,s) s> 0 1 { | } 2 { | } 0 (i, s ) (i = 1, 2, 0 For each t T , define the fundamental structure ∈ [ (H ) B(H ) B(H )] C t ⊆ t ⊆ t where H = L2(R2) ( t T ). Let u H = L2(R2) be a initial wave-function such that (k > 0, t ∀ ∈ 0 ∈ 0 0 small σ> 0): 1 x2 1 y2 u0(x,y) ψx(x, 0)ψy(y, 0) = exp ik0x 2 exp 2 ≈ √π1/2σ − 2σ · √π1/2σ − 2σ 0 0 where the average momentum (p1,p2) is calculated by ~ ~ 0 0 ∂ψx(x, 0) ∂ψy(y, 0) ~ (p1,p2)= ψx(x, 0) dx, ψy(y, 0) dy = ( k0, 0) ZR · i∂x ZR · i∂y That is, we assume that the initial state of the particle P ( in Figures 2(1) and 2(2) ) is equal to u u . | 0ih 0| As mentioned in the above, consider two branches T1 and T2. Thus, concerning T1, we have the following Schr¨odinger equation: ∂ ~2 ∂2 ~2 ∂2 i~ ψ (x,y)= ψ (x,y), = + V (x,y) ∂t t H1 t H1 −2m ∂x2 − 2m ∂y2 1 Also, concerning T2, we have the following Schr¨odinger equation: ∂ ~2 ∂2 ~2 ∂2 i~ ψ (x,y)= ψ (x,y), = + V (x,y) ∂t t H2 t H2 −2m ∂x2 − 2m ∂y2 2 Let s ,s be sufficiently large positive numbers. Put t = (1,s ) T , t = (2,s ) T . Define 1 2 1 1 ∈ 1 2 2 ∈ 2 the subtree T ′( T ) such that T ′ = 0,t1,t2 and 0 (x,y) R2 b x, δ(n 1) (E ) The measured date (x ,x . ,x ) Zk obtained by the parallel measurement K M 1 1 2 · · · K ∈ ⊗k=1 B(H0) 0,t1 O (Φ1 t1 ,S[ρ0]) (cf. [14]) will show the interference fringes. 0,t2 O Z Z 0,t2 Also, since Φ2 t2 = ( , 2 , Φ2 F ) is the observable in B(H0), we have the measurement M 0,t2 O B(H0)(Φ2 t2 ,S[ρ0]) (6) Fig. 2(2) says that M 0,t2 O (E2) if we get the positive measured value n by the measurement B(H0)(Φ2 t2 ,S[ρ0]), we may conclude that the particle P passed through the hole A. Further, note that we have the sequential causal observable [O ] = [ O , Φ0,ti : B(H ) T ′ { ti }i=1,2 { i ti → B(H ) ]. However, it should be noted that 0 }i=1,2 O (E3) the sequential causal observable [ T ′ ] can not be realized, since the commutativity does not generally hold, that is, it generally holds that Z Φ0,t1 F (Ξ) Φ0,t2 F (Γ) =Φ0,t2 F (Γ) Φ0,t1 F (Ξ) ( Ξ, Γ 2 ) 1 · 2 6 2 · 1 ∀ ∈ Remark 6 Although, strictly speaking, we have to say that the statement ”the particle P passed through the hole A” can not be described in terms of quantum language, it should be allowed to say the statement (E2). Also, concerning the statement (E3), note that O Z Z O t1 = ( , 2 , F )= t2 , O O but the observables t1 and t2 are in different worlds (i.e., different branches), except while 0,t1 0,t2 Φ1 = Φ2 . We consider that, the double-slit experiment can not be completely explained without branches In this sense, our argument may be similar to Everett’s (cf. [5]). Also, for our other understanding of the double-slit experiment, see [8] and [9]. 3 The quantum eraser experiment 3.1 Usual situation Let H be a Hilbert space. And let O = (X, , F ) be an observable in B(H). Let u and u ( H) F 1 2 ∈ be orthonormal elements, i.e., u = u = 1 and u , u = 0. Put k 1kH k 2kH h 1 2i u = α1u1 + α2u2 where α C such that α 2 + α 2 = 1. i ∈ | 1| | 2| Consider the measurement: MB(H)(O,S[ u u ]). (7) | ih | Then, the probability that a measured value x( X) belongs to Ξ( ) is given by ∈ ∈ F u, F (Ξ)u h i = α u + α u , F (Ξ)(α u + α u ) h 1 1 2 2 1 1 2 2 i = α 2 u , F (Ξ)u + α 2 u , F (Ξ)u + α α u , F (Ξ)u + α α u , F (Ξ)u | 1| h 1 1i | 2| h 2 2i 1 2h 1 2i 1 2h 2 1i = α 2 u , F (Ξ)u + α 2 u , F (Ξ)u + 2[Real part](α α u , F (Ξ)u ) | 1| h 1 1i | 2| h 2 2i 1 2h 1 2i where the interference term (i.e., the third term) appears. 9 3.2 Tensor Hilbert space 2 2 z1 Let C be the two dimensional Hilbert space, i,e., C = z1, z2 C . And put n z2 | ∈ o 1 0 e = , e = 1 0 2 1 Here, define the observable O = ( 1, 1 , 2 1,1 , F ) in B(C2) such that x {− } {− } x 1 1 1 1 1 1 F ( 1 )= , F ( 1 )= − , x { } 2 1 1 x {− } 2 1 1 − Here, note that 1 1 F ( 1 )e = (e + e ), F ( 1 )e = (e + e ) x { } 1 2 1 2 x { } 2 2 1 2 1 1 F ( 1 )e = (e e ), F ( 1 )e = ( e + e ) x {− } 1 2 1 − 2 x {− } 2 2 − 1 2 Also, define the existence observable O = ( 1 , 2 1 , F ) in B(C2) such that E { } { } E 1 0 F ( 1 )= E { } 0 1 Further, define ψ C2 H ( the tensor Hilbert space of C2 and H) such that ∈ ⊗ ψ = α e u + α e u 1 1 ⊗ 1 2 2 ⊗ 2 where α C such that α 2 + α 2 = 1. i ∈ | 1| | 2| 3.3 No interference Consider the measurement: M O O B(C2 H)( E ,S[ ψ ψ ]) (8) ⊗ ⊗ | ih | Then, we see (F ) the probability that a measured value (1,x)( 1 X) belongs to 1 Ξ is given by 1 ∈ { }× { }× ψ, (I F (Ξ))ψ h ⊗ i = α e u + α e u , (I F (Ξ))(α e u + α e u ) h 1 1 ⊗ 1 2 2 ⊗ 2 ⊗ 1 1 ⊗ 1 2 2 ⊗ 2 i = α e u + α e u , α e F (Ξ)u + α e F (Ξ)u h 1 1 ⊗ 1 2 2 ⊗ 2 1 1 ⊗ 1 2 2 ⊗ 2i = α 2 u , F (Ξ)u + α 2 u , F (Ξ)u | 1| h 1 1i | 2| h 2 2i where the interference term disappears. 10 3.4 Interference Consider the measurement: M O O B(C2 H)( x ,S[ ψ ψ ]) (9) ⊗ ⊗ | ih | Then, we see: (F ) the probability that a measured value (1,x)( 1, 1 X) belongs to 1 Ξ is given by 2 ∈ {− }× { }× ψ, (F ( 1 ) F (Ξ))ψ h x { } ⊗ i = α e u + α e u , (F ( 1 F (Ξ)))(α e u + α e u ) h 1 1 ⊗ 1 2 2 ⊗ 2 x { }⊗ 1 1 ⊗ 1 2 2 ⊗ 2 i 1 = α e u + α e u , α (e + e ) F (Ξ)u + α (e + e ) F (Ξ)u 2h 1 1 ⊗ 1 2 2 ⊗ 2 1 1 2 ⊗ 1 2 1 2 ⊗ 2i 1 2 2 = α1 u1, F (Ξ)u1 + α2 u2, F (Ξ)u2 + α1α2 u1, F (Ξ)u2 + α1α2 u2, F (Ξ)u1 2| | h i | | h i h i h i 1 2 2 = α1 u1, F (Ξ)u1 + α2 u2, F (Ξ)u2 + 2[Real part](α1α2 u1, F (Ξ)u2 ) 2| | h i | | h i h i where the interference term (i.e., the third term) appears. Also, we see: (F ) the probability that a measured value ( 1,x)( 1, 1 X) belongs to 1 Ξ is given 3 − ∈ {− }× {− }× by ψ, (F ( 1 ) F (Ξ))ψ h x {− } ⊗ i = α e u + α e u , (F ( 1 F (Ξ)))(α e u + α e u ) h 1 1 ⊗ 1 2 2 ⊗ 2 x {− }⊗ 1 1 ⊗ 1 2 2 ⊗ 2 i 1 = α e u + α e u , α (e e ) F (Ξ)u + α ( e + e ) F (Ξ)u 2h 1 1 ⊗ 1 2 2 ⊗ 2 1 1 − 2 ⊗ 1 2 − 1 2 ⊗ 2i 1 2 2 = α1 u1, F (Ξ)u1 + α2 u2, F (Ξ)u2 α1α2 u1, F (Ξ)u2 α1α2 u2, F (Ξ)u1 2| | h i | | h i− h i− h i 1 2 2 = α1 u1, F (Ξ)u1 + α2 u2, F (Ξ)u2 2[Real part](α1α2 u1, F (Ξ)u2 ) 2| | h i | | h i− h i where the interference term (i.e., the third term) appears. Remark 7 Note that (F1) = (F2)+(F3) no interference interferences are canceled This was experimentally examined in [27]. 4 Wheeler’s delayed choice experiment Let H be a two dimensional Hilbert space, i.e., H = C2. Let f ,f H such that 1 2 ∈ 1 0 f = , f = 1 0 2 1 Put f + f u = 1 2 √2 11 Thus, we have the state ρ = u u ( Sp(B(C2))). | ih | ∈ Let U( B(C2)) be an unitary operator such that ∈ 1 0 U = 0 eiπ/2 and let Φ : B(C2) B(C2) be the homomorphism such that → 2 Φ(F )= U ∗FU ( F B(C )) ∀ ∈ Consider two observable O = ( 1, 2 , 2 1,2 , F ) and O = ( 1, 2 , 2 1,2 , G) in B(C2) such that f { } { } g { } { } F ( 1 )= f f , F ( 2 )= f f { } | 1ih 1| { } | 2ih 2| and G( 1 )= g g , G( 2 )= g g { } | 1ih 1| { } | 2ih 2| where f1 + f2 f1 f2 g1 = , g2 = − √2 √2 half mirror 1 1 1 2 1 mirror 2 (f +f ) f √ √2 1 ✲ −−−−−−−→ √ 1 1 − f f √2 2 √2 1 √ 1 √ 1 ❄ − f2 − f2 √2 √2 ✲ D (= ( f f )) 2 | 2ih 2| mirror (photon counter) 1 f √2 1 ❄ D1(= ( f1 f1 )) (photon| counter)ih | Figure 3(1). [D1 + D2]=ObservableOf Firstly, consider the measurement: M O B(C2)(Φ f ,S[ρ]) (10) Then, we see: a measured value 1 (G ) the probability that is obtained by M C2 (ΦO ,S ) is given by 1 a measured value 2 B( ) f [ρ] tr(ρ ΦF ( 1 )) Uu, F ( 1 )Uu Uu, f 2 1 · { } = h { } i = |h 1i| = 2 tr(ρ ΦF ( 2 )) Uu, F ( 2 )Uu Uu, f 2 1 · { } h { } i |h 2i| 2 12 Next, consider the measurement: M 2 2O B(C )(Φ g,S[ρ]) (11) half mirror 1 1 1 2 1 mirror 2 (f +f ) f √ √2 1 ✲ −−−−−−−→ √ 1 1 − f f √2 2 √2 1 √ 1 ❄ − f2 0 √2 ✲ D (= ( g g )) 2 | 1ih 1| mirror half mirror 2 (photon counter) 1 f 1 f √2 1 − √2 2 ❄ D1(= ( g2 g2 )) (photon| counter)ih | Figure 3(2). [D1 + D2]=ObservableOg Then, we see: a measured value 1 2 (G ) the probability that is obtained by M C2 (Φ O ,S ) is given by 2 a measured value 2 B( ) g [ρ] tr(ρ Φ2G( 1 )) u, Φ2G( 1 )u u, UUg 2 0 · { } = h { } i = |h 1i| = tr(ρ Φ2G( 2 )) u, Φ2G( 2 )u u, UUg 2 1 · { } h { } i |h 2i| Also, consider the following Figure 3(3). This is clearly the same as the situation of Figure 3(1). M O Therefore, this is characterized by the same measurement B(C2)(Φ f ,S[ρ]). half mirror 1 1 1 2 1 2 (f +f ) f √ √2 1 ✲ D2(= ( f1 f1 )) −−−−−−−→ (photon| counter)ih | √ 1 − f √2 2 √ 1 ❄ − f √2 2 ✲ mirror half mirror 2 1 f √−2 2 ❄ D1(= ( f2 f2 )) (photon| counter)ih | Figure 3(3). [D2 + D1]=ObservableOf Remark 8. When the half mirror 2 is set in Figure 3(1)(i.e., when the observable Of changes to the O M 2 O M 2 2O g ), we see that the measurement B(C )(Φ f ,S[ρ]) changes to B(C )(Φ g,S[ρ]). Thus, we think 13 that Wheeler’s delayed choice experiment (cf. [28]) is not surprising in the linguistic interpretation of quantum mechanics. That is because the problem is not ”wave or particle” but ”interference or no M 2 O interference”. On the other hand, the statement (C1) concerning B(C )(Φ f ,S[ρ]) is surprising, since it implies the non-locality. This surprising fact is essentially the same as the de Broglie’s paradox (in B(L2(R3))). 5 Hardy’s paradox Let H be a two dimensional Hilbert space, i.e., H = C2. Let f ,f , g , g H such that 1 2 1 2 ∈ 1 0 f1 + f2 f1 f2 f1 = , f2 = , g1 = , g2 = − 0 1 √2 √2 Put f1 + f2 u = = g1 √2 Now, consider the tensor Hilbert space H H = C2 C2. Thus, put ⊗ ⊗ u = u u, ρ = u u u u ⊗ | ⊗ ih ⊗ | Define the projection P : C2 Cb2 C2 C2 suchb that ⊗ → ⊗ P (α f f + α f f + α f f + α f f )= α f f + α f f + α f f 11 1 ⊗ 1 12 1 ⊗ 2 21 2 ⊗ 1 22 2 ⊗ 2 12 1 ⊗ 2 21 2 ⊗ 1 22 2 ⊗ 2 and thus, define the Ψ : B(C2 C2) B(C2 C2) by ⊗ → ⊗ Ψ(A)= P AP (A B(C2 C2)) ∈ ⊗ b b b b 5.1 Concerning the tensor observable O O g ⊗ g Define the observable O = ( 1, 2 1, 2 , 2 1,2 1,2 , H ) in B(C2 C2) by the tensor observable gg { }×{ } { }×{ } gg ⊗ O O , that is, g ⊗ g b b H ( (1, 1) ) = g g g g , H ( (1, 2) ) = g g g g , gg { } | 1 ⊗ 1ih 1 ⊗ 1| gg { } | 1 ⊗ 2ih 1 ⊗ 2| Hbgg( (2, 1) ) = g2 g1 g2 g1 , Hbgg( (2, 2) ) = g2 g2 g2 g2 { } | ⊗ ih ⊗ | { } | ⊗ ih ⊗ | Consider theb measurement: b M O B(C2 C2)(Ψ gg,S[ρb]) (12) ⊗ b b M O Then, the probability that a measured value (2, 2) is obtained by B(C2 C2)(Ψ ,S[ρb]) is given by ⊗ b b u u, P H ( (2, 2) )P (u u) h ⊗ gg { } ⊗ i (f f ) (f f ),f f + f f + f f 2 =|h 1 − 2b⊗ 1 − 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| 16 f f f f f f + f f , f f + f f + f f 2 1 =|h 1 ⊗ 1 − 1 ⊗ 2 − 2 ⊗ 1 2 ⊗ 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| = 16 16 14 M O Also, the probability that a measured value (1, 1) is obtained by B(C2 C2)(Ψ gg,S[ρb]) is given by ⊗ u u, P H ( (1, 1) )P (u u) b b h ⊗ gg { } ⊗ i (f + f ) (f + f ),f f + f f + f f 2 =|h 1 2b⊗ 1 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| 16 f f + f f + f f + f f , f f + f f + f f 2 9 =|h 1 ⊗ 1 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| = 16 16 M O Further, the probability that a measured value (1, 2) is obtained by B(C2 C2)(Ψ gg,S[ρb]) is given by ⊗ b b u u, P H ( (1, 2) )P (u u) h ⊗ gg { } ⊗ i (f + f ) (f f ),f f + f f + f f 2 =|h 1 2b⊗ 1 − 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| 16 f f f f + f f f f , f f + f f + f f 2 1 =|h 1 ⊗ 1 − 1 ⊗ 2 2 ⊗ 1 − 2 ⊗ 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| = 16 16 Similarly, 1 u u, P H ( (2, 1) )P (u u) = h ⊗ gg { } ⊗ i 16 Remark 9. Recallingb Remark 1, note that 1 9 1 1 3 + + + = < 1 16 16 16 16 4 M O Thus, the probability that no measured value is obtained by the measurement B(C2 C2)(Ψ ,S[ρb]) ⊗ is equal to 1 . 4 b b 5.2 Concerning the tensor observable O O g ⊗ f Define the observable O = ( 1, 2 1, 2 , 2 1,2 1,2 , H ) in B(C2 C2) by the tensor observable gf { }×{ } { }×{ } gf ⊗ O O , that is, g ⊗ f b b H ( (1, 1) ) = g f g f , H ( (1, 2) ) = g f g f , gf { } | 1 ⊗ 1ih 1 ⊗ 1| gf { } | 1 ⊗ 2ih 1 ⊗ 2| Hbgf ( (2, 1) ) = g2 f1 g2 f1 , Hbgf ( (2, 2) ) = g2 f2 g2 f2 { } | ⊗ ih ⊗ | { } | ⊗ ih ⊗ | Consider theb measurement: b M O B(C2 C2)(Ψ gf ,S[ρb]) (13) ⊗ b b M O Then, the probability that a measured value (2, 2) is obtained by B(C2 C2)(Ψ gf ,S[ρb]) is given by ⊗ u u, P H ( (2, 2) )P (u u) b b h ⊗ gf { } ⊗ i (f f ) f ,f f + f f + f f 2 =|h 1 − 2b⊗ 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| = 0 8 M O Also, the probability that a measured value (1, 1) is obtained by B(C2 C2)(Ψ gf ,S[ρb]) is given by ⊗ u u, P H ( (1, 1) )P (u u) b b h ⊗ gf { } ⊗ i (f + f ) f ,f f + f f + f f 2 1 =|h 1 2b⊗ 1 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| = 8 8 15 M O Further, the probability that a measured value (1, 2) is obtained by B(C2 C2)(Ψ gf ,S[ρb]) is given by ⊗ b b u u, P H ( (1, 2) )P (u u) h ⊗ gf { } ⊗ i (f + f ) f ,f f + f f + f f 2 4 =|h 1 2b⊗ 2 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| = 16 8 Similarly, u u, P H ( (2, 1) )P (u u) h ⊗ gf { } ⊗ i (f f ) f ,f f + f f + f f 2 1 =|h 1 − 2b⊗ 1 1 ⊗ 2 2 ⊗ 1 2 ⊗ 2i| = 8 8 Remark 10. It is usual to consider that ”Which way pass problem” is nonsense. However, for the other aspect of this problem, see Remarks 11 and 12 later. 6 The three boxes paradox Let H be the three dimensional Hilbert space, i.e., H = C3. Let f ,f ,f H such that 1 2 3 ∈ 1 0 0 f1 = 0 , f2 = 1 , f3 = 0 0 0 1 Put f1 + f2 + f3 f1 + f2 f3 u = , g1 = − . √3 √3 And, put ρ = u u | ih | Further, consider two observables O = ( 1, 2 , 2 1,2 , G) and O = ( 1, 2, 3 , 2 1,2,3 , F ) in B(H) 1 { } { } 2 { } { } such that 1 G( 1 )= g g = f + f f f + f f ( P ), G( 2 )= I g g ( P ) { } | 1ih 1| 3| 1 2 − 3ih 1 2 − 3| ≡ 1 { } − | 1ih 1| ≡ 2 and F ( 1 )= f f , F ( 3 )= f f , F ( 3 )= f f , { } | 1ih 1| { } | 2ih 2| { } | 3ih 3| And consider the measurements MB(H)(O1,S[ρ]) and MB(H)(O2,S[ρ]) (14) Clearly, the probability that a measured value 1 obtained by MB(H)(O1,S[ρ]) is given by 1 1 u, G( 1 )u = u, g g u = f + f + f ,f + f f 2 = h { } i h | 1ih 1| i 9|h 1 2 3 1 2 − 3i| 9 measured value 1 M O and, the probability that a measured value 2 obtained by B(H)( 2,S[ρ]) is given by measured value 3 2 1 u, f1 f1 u = f1, u = 3 h | ih | i |h i|2 1 u, f2 f2 u = f2, u = h | ih | i |h i| 3 u, f f u = f , u 2 = 1 h | 3ih 3| i |h 3 i| 3 16 Since O and O do not commute, the simultaneous observable O O does not exist. However, 1 2 1 × 2 putting X = 1, 2 , Y = 1, 2, 3 , O O may be formally written by { } { } 1 × 2 X Y O O = (X Y, 2 × , G F ) 1 × 2 × × where 1 G( 1 )F ( 1 ) = ( g ,f ) g f = f + f f f { } { } h 1 1i | 1ih 1| 3| 1 2 − 3ih 1| 1 G( 1 )F ( 2 ) = ( g ,f ) g f = f + f f f { } { } h 1 2i | 1ih 2| 3| 1 2 − 3ih 2| 1 G( 1 )F ( 3 ) = ( g ,f ) g f = f + f f f { } { } h 1 3i | 1ih 3| 3| 1 2 − 3ih 3| ······ However, we try to consider the ”measurement” ”M (O O ,S )”. (15) B(H) 1 × 2 [ρ] And further, we can calculate as follows. (H) under the condition that the measured value (1,y) is obtained by ”M (O O O ,S )”, B(H) 1 × 2 × 2 [ρ] y = 1 the probability that y = 2 is formally given by y = 3 u,G( 1 )F ( 1 )u u,( g1,f1 ) g1 f1 u g1, f1 f1 u h { } { } i h h i | ih | i h | ih | i u,G( 1 )u = u, g1 g1 u = g1 u = 1 u,Gh ( 1 ){F}( 2i )u u,( hg1,f| 2 ih) g1| if2 u g1,hf2 | fi2 u h { } { } i h h i | ih | i h | ih | i u,G( 1 )u = u, g1 g1 u = g1 u = 1 u,G(h 1 )F{ (}3 i)u u,( gh1,f|3 )ihg1 |fi3 u g1, fh3 |f3i u h { } { } i h h i | ih | i h | ih | i u,G( 1 )u = u, g1 g1 u = g1 u = 1 h { } i h | ih | i h | i − which shows the strange fact (i.e., ”minus probability”). Remark 11. Since O and O do not commute, O O is not an observable, but B(H)-valued 1 2 1 × 2 measure space (cf. Remark 5). Thus, it is usual to consider that the above (H) is meaningless. However, if some will find the idea such that the (H) becomes meaningful, then the idea should be added to the linguistic interpretation mentioned in Section 1.4. For example, the idea in ref. [1] (the weak value associated with a weak measurement) may be somewhat hopeful, but we can not assure it. Remark 12 (Continued from Hardy’s paradox in Section 5). Define the observable O = ( 1, 2 ff { }× 1, 2 , 2 1,2 1,2 , H ) in B(C2 C2) by the tensor observable O O , that is, { } { }×{ } ff ⊗ f ⊗ f b b H ( (1, 1) ) = f f f f , H ( (1, 2) ) = f f f f , ff { } | 1 ⊗ 1ih 1 ⊗ 1| ff { } | 1 ⊗ 2ih 1 ⊗ 2| Hbff ( (2, 1) ) = f2 f1 f2 f1 , Hbff ( (2, 2) ) = f2 f2 f2 f2 { } | ⊗ ih ⊗ | { } | ⊗ ih ⊗ | b b In spite of the non-commutativity of ΨOgg and Off , consider the ”measurement”: M b b Ob O B(C2 C2)(Ψ gg ff ,S[ρb]) (16) ⊗ × And we can calculate as follows. b b b 17 M O (I) under the condition that the measured value ((2, 2), (y1,y2)) is obtained by ” B(C2 C2)(Ψ gg ⊗ × y ,y , ( 1 2) = (1 1) b b (y1,y2) = (1, 2) O ,S b )”, the probability (or precisely, weak value) that is formally given ff [ρ] (y ,y ) = (2, 1) 1 2 b (y ,y ) = (2, 2) 1 2 by b b u u,P Hgg( (2,2) )P Hff ( (1,1) )(u u) h ⊗ { b } { } ⊗ i u u,P Hgg( (2,2) )P (u u) h b⊗ { b } ⊗ i u u,P Hgg( (2,2) )P Hff ( (1,2) )(u u) h ⊗ { b } { } ⊗ i u u,P Hgg( (2,2) )P (u u) h b⊗ { b } ⊗ i u u,P Hgg( (2,2) )P Hff ( (2,1) )(u u) h ⊗ { b } { } ⊗ i u u,P Hgg( (2,2) )P (u u) h b⊗ { b } ⊗ i u u,P Hgg( (2,2) )P Hff ( (2,2) )(u u) h ⊗ { b } { } ⊗ i u u,P Hgg( (2,2) )P (u u) h ⊗ { } ⊗ i (f1 + f2) (f1 + f2), P g2 g2 g2 g2 P f1 f1 f1 f1, (f1 + f2) (f1 + f2) h ⊗ | ⊗ ih ⊗ | | ⊗ ih ⊗ ⊗ i (f1 + f2) (f1 + f2), P g2 g2 g2 g2 P f1 f2 f1 f2, (f1 + f2) (f1 + f2) =4h ⊗ | ⊗ ih ⊗ | | ⊗ ih ⊗ ⊗ i (f1 + f2) (f1 + f2), P g2 g2 g2 g2 P f2 f1 f2 f1, (f1 + f2) (f1 + f2) h ⊗ | ⊗ ih ⊗ | | ⊗ ih ⊗ ⊗ i (f1 + f2) (f1 + f2), P g2 g2 g2 g2 P f2 f2 f2 f2, (f1 + f2) (f1 + f2) h ⊗ | ⊗ ih ⊗ | | ⊗ ih ⊗ ⊗ i (f1 + f2) (f1 + f2), f1 f2 f2 f1 + f2 f2 f1 f2 f2 f1 + f2 f2,f1 f1 h ⊗ − ⊗ − ⊗ ⊗ ih− ⊗ − ⊗ ⊗ ⊗ i (f1 + f2) (f1 + f2), f1 f2 f2 f1 + f2 f2 f1 f2 f2 f1 + f2 f2,f1 f2 =h ⊗ − ⊗ − ⊗ ⊗ ih− ⊗ − ⊗ ⊗ ⊗ i (f1 + f2) (f1 + f2), f1 f2 f2 f1 + f2 f2 f1 f2 f2 f1 + f2 f2,f2 f1 h ⊗ − ⊗ − ⊗ ⊗ ih− ⊗ − ⊗ ⊗ ⊗ i (f1 + f2) (f1 + f2), f1 f2 f2 f1 + f2 f2 f1 f2 f2 f1 + f2 f2,f2 f2 h ⊗ − ⊗ − ⊗ ⊗ ih− ⊗ − ⊗ ⊗ ⊗ i 0 0 ( 1) ( 1) 1 = = (−1) × (−1) 1 −( 1)× −(1) 1 − × − This (I) and the idea in ref. [1] are superficially similar, but completely different in essence. How- ever, if the latter says something good, we can expect that the (I) is somewhat meaningful. For completeness, note that quantum language is not physics but language. Therefore, we say that (J) if this statement (I) can be used effectively, then the concept: ”weak value” should be accepted in the linguistic interpretation, however, if this is not more than ”even not wrong”, we will not be concerned with ”the weak value”. 7 Conclusions In this paper, we discussed the double slits experiment, the quantum eraser experiment, Wheeler’s delayed choice experiment, Hardy’s paradox and the three boxes paradox in the linguistic interpre- tation of quantum mechanics. Quantum language says that everything should be described in terms of Axioms 1 and 2. There- fore, we always have to describe ”measurement” explicitly. In fact, in this paper, any measurement was explicitly described such as the formula [(5)-(14), (15), (16) ]. Particularly, in Section 2, we say that the double-slit experiment can not be understood without the concept of ”branch”. And, in Section 4, we note that Wheeler’s delayed choice experiment is not surprising, since it should be regarded as the problem such as ”interference or no interference”. Through these arguments, we assert that the linguistic interpretation is just the final version of so called Copenhagen interpretation. And therefore, the Copenhagen interpretation does not belong to physics but the linguistic world view (cf. Figure 1). We hope that our proposal will be discussed and examined from various view-points. 18 References [1] Y. Aharonov, D. Z. Albert and L. Vaidman, How the Result of a Measurement of a Component of a spin-1/2 Particle Can Turn Out to be 100, Phys Rev. Lett.,60, 1351-1354, 1988 DOI: http://dx.doi.org/10.1103/PhysRevLett.60.1351 [2] J. S. Bell, “On the Einstein-Podolosky-Rosen Paradox,” Physics, Vol. 1, 1966, 195-200. [3] E. B. Davies, Quantum Theory of Open Systems, Academic Press, 1976 [4] A. Einstein, B. Podolosky and N. Rosen, “Can Quantum-mechanical Description of Physical Reality be Considered Complete?” Physical Review, Vol. 47, No. 10, 1935, pp. 777-780. doi: 10.1103/PhysRev.47.777 [5] H. Everett ’Relative stat’ Formulation of Quantum mechanics, Review of Modern Physics 29 1957 [6] R. P. Feynman The Feynman lectures on Physics; Quantum mechanics Addison-Wesley Publishing Company, 1965 [7] L. Hardy, ”Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories, Physical Review Letters 68 (20): 2981-2984 1992 DOI: http://dx.doi.org/10.1103/PhysRevLett.68.2981 [8] S. Ishikawa, Uncertainty relation in simultaneous measurements for arbitrary observables, Rep. Math. Phys., 9, 257-273, 1991 doi: 10.1016/0034-4877(91)90046-P [9] S. Ishikawa, T. Arai, T. Kawai, Numerical Analysis of Trajectories of a Quantum Particle in Two-slit Experiment, International Journal of Theoretical Physics, Vol. 33, No. 6, 1265-1274, 1994 doi: 10.1007/BF00670793 [10] S. Ishikawa, A Quantum Mechanical Approach to Fuzzy Theory, Fuzzy Sets and Systems, Vol. 90, No. 3, 277-306, 1997 doi: 10.1016/S0165-0114(96)00114-5 [11] S. Ishikawa, T. Arai, T. Takamura, A dynamical system theoretical approach to Newtonian mechanics, Far east journal of dynamical systems 1, 1-34 (1999) (http://www.pphmj.com/abstract/191.htm) [12] S. Ishikawa, Statistics in measurements, Fuzzy sets and systems, Vol. 116, No. 2, 141-154, 2000 doi:10.1016/S0165-0114(98)00280-2 [13] S. Ishikawa, Mathematical Foundations of Measurement Theory, Keio University Press Inc. 335pages, 2006. (http://www.keio-up.co.jp/kup/mfomt/) [14] S. Ishikawa, A New Interpretation of Quantum Mechanics, Journal of quantum information science, Vol. 1, No. 2, 35-42, 2011 doi: 10.4236/jqis.2011.12005 (http://www.scirp.org/journal/PaperInformation.aspx?paperID=7610) [15] S. Ishikawa, The linguistic interpretation of quantum mechanics, arXiv:1204.3892v1[physics.hist-ph] , (2012), (http://arxiv.org/abs/1204.3892) [16] S. Ishikawa, Quantum Mechanics and the Philosophy of Language: Reconsideration of traditional philosophies, Journal of quantum information science, Vol. 2, No. 1, 2-9, 2012 doi: 10.4236/jqis.2012.21002 (http://www.scirp.org/journal/PaperInformation.aspx?paperID=18194) [17] S. Ishikawa, A Measurement Theoretical Foundation of Statistics, Journal of Applied Mathematics, Vol. 3, No. 3, 283-292, 2012 doi: 10.4236/am.2012.33044 (http://www.scirp.org/journal/PaperInformation.aspx?paperID=18109&) 19 [18] S. Ishikawa, Measurement theory in the philosophy of science, arXiv:1209.3483 [physics.hist-ph] 2012. (http://arxiv.org/abs/1209.3483) [19] S. Ishikawa, A quantum linguistic characterization of the reverse relation between confidence interval and hypothesis testing, arXiv:1401.2709v2 [math.ST] 2014. (http://arxiv-web3.library.cornell.edu/abs/1401.2709v2) [20] S. Ishikawa, Heisenberg uncertainty principle and quantum Zeno effects in the linguistic interpretation of quantum mechanics, arXiv:1308.5469 [quant-ph] 2014. (http://arxiv.org/abs/1308.5469) [21] S. Ishikawa, Regression analysis in quantum language, arXiv:1403.0060 [math.ST] 2014. (http://arxiv.org/abs/1403.0060) [22] S. Ishikawa, Kalman filter in quantum language, arXiv:1404.2664 [math.ST] [math.ST] 2014. (http://arxiv.org/abs/1404.2664) [23] A. Kolmogorov, Foundations of probability ( translation ), Chelsea Publishing Co. 1950 [24] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Springer Verlag, Berlin, 1932 [25] T. Ravon and L. Vaidman, “ The three-box paradox revisited,” J. Phys. A: Math. Theor. 40 (2007) 2873-2882 (http://iopscience.iop.org/1751-8121/40/11/021/) [26] S. Sakai, C∗-algebras and W ∗-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete (Band 60), Springer-Verlag, (1971) [27] S. P.Walborn, et al. “Double-Slit Quantum Eraser,” Phys.Rev.A 65, (3), 2002 DOI: http://dx.doi.org/10.1103/PhysRevA.65.033818 [28] J. A. Wheeler, ”The ’Past’ and the ’Delayed-Choice Double-Slit Experiment’,” pp 9-48, in A.R. Marlow, editor, Mathematical Foundations of Quantum Theory, Academic Press (1978)