 Journal of Quantum Informatio n Science, 2011, 1, 35-42 doi:10.4236/jqis.2011.12005 Published Online September 2011 (http://www.SciRP.org/journal/jqis) Copyright © 2011 SciRes. JQIS A New Interpretation of Quantum Mechanics Shiro Ishikawa Department of Mat hematics, Faculty of Science an d Te ch n ol ogy, Keio University, Hiyoshi, Yokohama, Japan E-mail: ishikawa@math.keio.ac.jp Received July 17, 2011; revised August 1, 2011; accepted August 15, 2011 Abstract The Copenhagen interpretation is the most authorized interpretation of quantum mechanics, but there are a number of ideas that are associated with the Copenhagen interpretation. It is certain that this fact is not nec- essarily desirable. Thus, we propose a new interpretation of measurement theory, which is the linguistic as- pect (or, the mathematical generalization) of quantum mechanics. Although this interpretation is superficially similar to a part of so-called Copenhagen interpretation, we show that it has a merit to be applicable to both quantum and classical systems. For example, we say that Bell’s inequality is broken even in classical sys- tems. Keywords: the Copenhagen Interpretation, Quantum and Classical Measurement Theory, the Law of Large Numbers, Maximum Likelihood Estimation, Kolmogorov Extension Theorem, Wavefunction Collapse, Bell’s Inequality 1. Introduction It is well known (cf. [1]) that quantum mechanics is for- mulated in an operator algebra (i.e., an operator algebra composed of all bounded linear operators on a Hilbert space H with the norm BH = BH F=1 sup uH u) as follows: physics probabilistic interpretationkinetic equation quantum mechanics quantum measurementcausality A Also, the Copenhagen interpretation due to N. Bohr (et al.) is characterized as the guide to the usage of quantum mechanics (A). Although quantum mechanics (A) with the Copenhagen interpretation is generally accepted as one of the most trustworthy theories in science, it should be noted that there is no definitive statement of the Co- penhagen interpretation, that is, there are a number of ideas that are associated with the Copenhagen interpreta- tion. We do not think that this fact is desirable. Measurement theory (mentioned in Section 2 later or refs. [2-6]) is, by an analogy of the (A), constructed as the mathematical theory formulated in a certain C*-alge- bra A (i.e., a norm closed subalgebra in , cf. [7]) as follows: BH language measurement theoryB Rule 1 in Section 2Rule 2 in Section 2 measurementcausality Note that this theory (B) is not physics but a kind of language based on the mechanical world view since it is a mathematical generalization of quantum mechanics (A). When =c BH , the C*-algebra composed of all compact operators on a Hilbert space H, the (B) is called quantum measurement theory (or, quantum system the- ory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when is commutative (that is, when is characterized by , the C*-algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space 0 C (cf. [7])), the (B) is called classical measure- ment theory. Thus, we have the following classification: 0 when when measurement theory quantum measurement theory =classical measurement theory c BH C C That is, this theory covers several conventional system theories (i.e., statistics, dynamical system theory, quan- tum system theory). The purpose of this paper is to propose an interpreta- tion of measurement theory (B). Since the (C) says that this interpretation should be common in classical and  36 S. ISHIKAWA quantum measurement theories, it is also regarded as a new interpretation of quantum mechanics. 2. Measurement Theory Now we shall explain the measurement theory (B). Let be a -algebra, and let be the dual Banach space of . That is, BH* C is a continu- ous linear functional on , and the norm is defined by () sup {suchthat=BH FF F F 1. Define the mixed state such that =1 and And define the mixed state space such that 0 for allthat0.FF F such m S is a mixed state. m S 2 (1) A mixed state ) is called a pure state if it satisfies that m S 1 =1 for some 12 , and 0< m S <1 implies 1 == 2 . Put is a pure state. pm SS (2) which is called a state space. It is well known (cf. [7]) that * ()= pc BH uuS (i.e., the Dirac notation) =1 H u, and * 000 pC S is a point measure at , where 0 0 0dff . The latter implies that 0 fC * 0 pCS can be also identified with such as * 00 0 pC S∋ * . Here, assume that the -algebra C BH has the identity . This assumption is not unnatural, since, if , it suffices to reconstruct the above such that it includes I . According to the noted idea (cf. [8]) in quantum mechanics, an observable O:= X, , in is defined as follows: 1) [Field] is a set, X, the power set of is a field of , that is, “12 ,1 2 ”, “”. \X 2) [Finite additivity] is a mapping from to satisfying: a): for every , is a non- negative element in such that F 0 I , b): and =0F = XI , where 0 and I is the 0-ele- ment and the identity in respectively. c): for any finite decomposition of 12 ,,,,, n N , it holds that ..,,(= 1,ieF n =ij =1 2,3,,),= , N nn N N n FF n ij =1 n . Remark 1 [Countable additivity] The assumption of the countable additivity (i.e., =1 lim N Nn n F F in the sense of weak-topology ) may be rather half-fin- ished. If the countable additivity is required, it is, by the reason mentioned in Remark 3 and 6 later, recommended to start from the -algebra * W such that BH as discussed in Appendix later (cf. [5]). However, our interest in this paper is not mathematics but the in- terpretation of measurement theory. Thus, all arguments will be discussed under the above finite additivity (i.e., 1) and 2)). With any system S, a -algebra * C BH can be associated in which the measurement theory (B) of that system can be formulated. A state of the system S is represented by an element and an ob- servable is represented by an observable p S =, ,O: F in . Also, the measurement of the observable for O the system S with the state is denoted by MO,S (or more precisely, MO:=(,,), FS ). An ob- server can obtain a measured value X by the measurement MO,S The Rule 1 presented below is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics (A). And thus, it is a statement without reality. . Rule 1 [Measurement] The probability that a meas- ured value X obtained by the measurement 0 MO:=(,,),XFS belongs to a set is given by 0F . Remark 2 Again note that Rule 1 is a statement with- out reality (i.e., a kind of incantation or spell). Thus, it is unnecessary (or precisely speaking, impossible) to an- swer the question: “What is measurement (or, system, state, observable, probability, etc.)?” However, surpris- ingly, as seen in [2-6] or Section 4 later, man’s linguistic competence enables us to use Rule 1. This is essential to our approach to the interpretation of quantum mechanics. Next, we explain Rule 2 in (B). Let 11 BH and 22 BH 1,2 : be -algebras. A continuous lin- ear operator * C 2 1 2 is called a Markov opera- tor, if it satisfies that 1): for any non- negative element 1,2 20F in , ii): 1,2 21 2 = I, where k is the identity in 1,2.,=k k Here note that, for any observable 22 O:=, , F in 2 the , 1,2 2 ,, F 1,2 2 O. is an observable in , which is denoted by 1 Also, the dual operator clearly satisfies that ** * 1,2 12 : * 1, 2 ** 12 mm SS. Let ,T be a tree, i.e., a partial ordered finite set such that 1 t3 t and 2 tt 3 implies or 1 tt221 tt . Assume that there exists an element , called the root of T, such that () holds. Put 0 tT tT 0 tt 2 tt 22 12 =,Ttt 1 T . C opyright © 2011 SciRes. JQIS  37 S. ISHIKAWA Definition 1 [Markov relation] The family 12 ,: tt is called a Markov relation (due to 21 12 2 (, ) tt tt T the Heisenberg picture), if it satisfies the following con- ditions 1) and 2). 1) With each , a -algebra is associated. tT ,tt * C 2 Tt 2) For every 12 , a Markov operator 1 t is defined. And it satisfies that holds for any , . 12 ,tt 12 ,: tt 12 23 ,,tt tt 2 t =tt 13 , The family of dual operators 2 23 ,ttT 1212 12 ** * ,2 (, ) :mm tttt tt T SS is called a Markov relation (due to the Schrödinger pic- ture). However, it is not formally used in measurement theory. Now Rule 2 in the measurement theory (B) is pre- sented as follows: Rule 2 [Causality] The causality is represented by a Markov relation . 1221 12 ,2 , : tttt ttT Remark 3 If an infinite tree T is required, we must start from a -algebra * W (cf. Appendix later). However, in this paper, we, for simplicity, assume the finiteness of . Also, by the same reason mentioned in Remark 2, the question: “What is causality?” is nonsense. What we can do in measurement theory is only to trust in man’s lin- guistic competence. T 3. An Interpretation of Measurement Theory The measurement theory (B) asserts “Describe any ordi- nary phenomenon according to Rules 1 and 2”. Still, most readers may be perplexed how to use Rules 1 and 2 since there are various usages. Thus, the following prob- lem is significant. (D) How should Rules 1 and 2 be used? Note that reality is not reliable since Rules 1 and 2 are statements without reality. Thus, we want to define the new interpretation such that (E) the new interpretation is a guide to the most useful (or, powerful, “Occam’s razor”-like) usage of Rules 1 and 2. Now we can present our main assertion in this paper as follows: (F) The new interpretation defined in the (E) is charac- terized as the following (F1) – (F3). Here, (F1) Consider the dualism composed of “observer” and “system (= measuring object)”. (F2) Only one measurement is permitted. And thus, the state after a measurement is meaningless since it can not be measured any longer. (F3) The causality should be assumed only in the side of system, however, a state never moves. Thus, the Heisenberg picture should be adopted. Also, the observer does not have the space-time. Thus, the question: “When and where is a measured value obtained?” is meaningless, and so on. The above may be rather similar to a certain part of so called Copenhagen interpretation. However, note that we do not assume “the state after a measurement (= wave- function collapse)” and “the Schrödinger picture”, which are often investigated in so-called Copenhagen interpre- tation. Also, some may consider that the above proposal (F) is too optimistic, since the following question is not yet answered: (G) Does the most useful usage of Rules 1 and 2 exist? Or, is it determined uniquely? However, we may be allowed to be optimistic until another most useful usage (or, a powerful rival candidate) will be discovered. In other words, we expect the readers to read the overwhelming predominance of the (F) in the following section or refs. [2-6]. 4. Examples and Remarks What we want to assert in this paper is only the (F). Thus, it is desirable that the each one reader verifies the supe- riority of the (F) in our papers [2-6]. However, in this section we take up some simple examples, which pro- mote the readers’ understanding of the (F). Note, for completeness, that all examples are consequences of measurement theory with the interpretation (F). Example 1 [Parallel measurement, the law of large numbers] For each , consider a measure- ment =1,2, ,k K MO:= [] ,,,XFS . k kkkk k However, the in- terpretation (F2) says that only one measurement is per- mitted. Thus, we consider the spatial tensor -algebra * C =1=1 , kk kk BH K KK and consider the product space =1kk and the product field =1 k , which is defined by the smallest field that contains a family K k =1 ,=1,2, , K kkkk kK . Define the parallel observable =1=1 =1 O, KKK kk kkkk , F *K in the tensor -algebra such that C=1k k =1 =1 =,= kk kkkk k KK 1,2,,. FkK Then, the above =1 MO, kk kk S is represented Copyright © 2011 SciRes. JQIS  38 S. ISHIKAWA by the parallel measurement =1 =1 =1 MO KK kk kk K kk S =1 MO, kk K kk S =k O= ,, =XF , which is also denoted by . Consider a particular case such that, , ,, , kkk F Let 1,2,,.kK 12 ,, ,= = k xx X =1 K k be a measured value by the parallel measure- ment . Then, using Rule 1, we see the law of large numbers, that is, for sufficiently large K, MO,S #1,2,, k kKx F #[ ] holds (cf. [2,4]), where is the the number of ele- ments of the set A. This is, of course, most fundamental in science. Also, this is the reason that the term “prob- ability” is used in Rule 1. Notation 1 It is natural to consider that a measurement is usually MO:=,, ,XFS MO,S taken in order to know the state . Thus, when we want to emphasize that we do not know the state , the me- asurement MO is often denoted by ,S MO, .S Remark 4 [Maximum likelihood estimation in classical measurements] Consider the classical cases in (C). It may be usual to consider that Rule 1 leads the following statement (cf. [3,4]), i.e., maximum likelihood estimation in classical measurements: (H) When we know that a measured value obtained by a measurement 1 , 0111 :=,, C MO FS 1 * 00 *pC S be- longs to , there is a reason to infer that the unknown state where 1 * 0 pC 0 S is defined by 011=F 11 F 0 if it exists. 0* pC S max Although this (H) is surely handy, note that the (F2) says that it is illegal to regard the as the state after the measurement . Thus, strictly speak- 0 C 1 MO,S ing, the (H) is informal. And thus, it should be reconsid- ered in Example 2 below. Example 2 [Maximum likelihood estimation in meas- urements] By a similar method as the lead of the (H), we can easily see the following statement (I), which should be regarded as the measurement theoretical form of ma- ximum likelihood estimation (cf. Corollary 5.5 in [4]). (I) When we know that a measured value obtained by a measurement , , 1212 [] MO:= , XF S 2 1 belongs to , there is a reason to infer that the probability that the measured value belongs to 12 22 is given by the following con- ditional probability: 01 01 F 2 2 FX (3) where p S 0 is defined by 012 FX 12 FX ma xp S if it exists. Here, note that the 0 is not the state after the measurement MO, .S This (I), which also includes quantum cases, is most fundamental in statistics, and thus, we believe (cf. [2-6]) that statistics is one of aspects of measurement theory. For the relation between the informal (H) and the formal (I), see Remark 5 later. Example 3 [Simultaneous measurement] For each , consider a measurement =1,2, ,kK ,, , kkkk XFS MO:= . However, since the (F2) says that only one measurement is permitted, the =1 ,MO kk S should be recon- sidered in what follows. Under the commutativity condi- tion such that ,, =, jjii j F F ij = ijj ii j FF FF i (4) we can define the simultaneous observable =1 =1 , KK kkk = O= KK kk 1 =1 , kkk F in such that =1 = ,= KK k k kk 11 22 1,,. K K FFFF kK =1kk (5) Then, the above =1 MO, kk S is, under the commutativity condition (4), represented by the simulta- neous measurement =1 MO, K kk S . Remark 5 [The relation between (H) and (I)] Consider the (I) in the classical cases, i.e., . And as- sume the simultaneous observable 12 0 =C = FF in (3). Then, putting 00 = (i.e., the point measure at 0 ), we see that 11 2 11 2 FF FF 2 0 22 0 2 0 =F X (3) 02 2 =F Since this equality holds for any 2222 O=,, F and any 22 , some may want to regard the 0 as the state after the measurement 01111 MO:=,,, C FS in the (H). Thus, in spite of the (F2), the (H) may be al- C opyright © 2011 SciRes. JQIS  39 S. ISHIKAWA lowed in classical cases if the 0 may be regarded as something represented by the term such as “imaginary state” (cf. [5]). This is the meaning of the informal (H). Example 4 [How to use Rule 2 (Causality)] Consider a tree 01 :=, ,,, n Ttt t with the root . This is also characterized by the map 0 t 0 π:Tt T such that π=max <tsTst. Let : tt 2 ,, tttT t be a Markov relation, which is also represented by 0 π,π :t ttttTt . Let an observable O: = t ,, ttt F * 0 pt S in the be given for each . And let . Consider “measurements” such as t T tT 0 (J) for each , take a measurement of an observ- able for the system with a “moving state” t Ot 0 ** * 0π,π ;: mm t tt ttT t SS , where the meaning of “moving state” is not clear yet. Recalling that the (F3) says that a state never moves, we consider the meaning of the (J) as follows: For each T, put = s TtTts . And define the observable O= , , stT t tT t ss F in (due to Heisenberg picture) as follows: 1π, π Oifπ O=O(O) ifsπ s s t stt ts TT T s (6) if the commutativity condition holds (i.e., if the simulta- neous observable exists) for 1 ππ, OO t ststt each π T. Using (6) iteratively, we can finally obtain the observable 0 t in0 t Thus the above (J) is O represented by the measurement . This also satisfies the (F2). 00 0[] MO, t tS Remark 6 [Kolmogorov extension theorem] In the general cases such that countable additivity and infinite (in Remarks 1 and 3) are required, the existence of the above T 0 Ot is, by using the Kolmogorov extension theorem in probability theory [9], proved in the - algebraic formulation (cf. [4,5] and Appendix later). We think that this fact is evidence that the interpretation (F2) is hidden behind the utility of the Kolmogorov extension theorem. Recall the following well-known statement that always appears in the beginning of probability theory: * W (K) Let ,, P P be a probability space. Then, the probability that an event occurs is given by , which, as well as Rule 1, is a statement without reality. We consider that the Kolmogorov extension theorem is regarded as one of the finest answers to the problem: How should the statement (K) be used? That is, in mathematical probability theory, the answer is presented as the form of a basic theorem (i.e., the Kolmogorov ex- tension theorem). On the other hand, in measurement theory, the problem (D) is answered by the interpretation (F). Remark 7 [Wavefunction collapse] Again reconsider the (J) in the simplest case that 011 0 =,, π=Tttt t ,S . Taking a measurement t, we know that the measured value belongs to . Then, it may 00 0 MO t 0 0t be usual to consider that a certain wavefunction collapse happens by the measurement, that is, *pt S∋ 0 0 * 0 00 . pt S And continuously, we take a mea- surement 011 00 0 , MO, tt t tS . Here, the probability that a measured value belongs to is, by Rule 1, given by 1 1t 0 01 0,11 . tt F However, this must be equal to the conditional 01 0,11tt F 0 probability 001 001 F FX if the commutativity condition holds (i.e., the simultane- ous observable 00011 0101 0011 , , O=O O =,,:= ttttt tttt tttt XFF F exists). This implies that it suffices to consider only the measurement 0011 0 0, MO O, At ttt tS . That is, two measurements 00 0 MO, t tS and 01 10 00 , MO, tt t tS are not needed. Also, if the commutativity condition is ignored in the argument of the wavefunction collapse, it is doubtful. Example 5 [Bell’s inequality] According to [11], we shall study the following steps [1-3] in measurement theory. [Step 1]: Put =1,1X. Let 12 =1a and 12 =1b be complex numbers such that a 22 12 =1 and 22 12 =1b . Consider a probability space 22 ,, ab XX such that 1122 112 2 1,1 =1,1 =14 1,1=1,1= 14. ab ab ab ab Define and calculate the correlation function ,Pab such that Copyright © 2011 SciRes. JQIS  40 S. ISHIKAWA 12 12121122 (,) ,,= ab xx XX Pabxxx x (7) Our problem is as follows. (L) Find a measurement in a C*-algebra such that 0 22 MO:= ,, , ab ab XXFS 0 = ab ab F 2.PX This will be answered in the following step [2]. [Step: 2]. In what follows, we shall investigate the (L) in two cases (i.e., quantum case 22 =B 2 . and classical case ). 0 =C 1): Quantum case [ ( cf. [10,11])] 222 ==BBB P ut 2 12 10 =,= 01 ee For each , define the observables ,cab O,, cc XG in such that 2 B 11 11 1,1 11 22 cc cc GG cc . Further consider the quantum system composed of two particles formulated in Put 22 .B = ss and * 22 000 =, pB S where = s 1221 2eeee and 01 =ee 1 22 . Consider the unitary operator UB such that 0=. U Define the observable 22 O,, ab ab XXF:= *ab UG GU in Thus we get a measurement 22 .B 22 0 MO, ab BS in . 22 B This clearly satisfies the (L) since we easily calculate that, for each 2 12 , xX, 012 0120 12 12 , =, , =, =,. ab ab ab ab Fxx Fxx Gx Gx xx 2): Classical case [ 000 ==AC C C ] Let 0 00 =, , and put 00 = * 0 ( pCS, And define the observable 22 O:= ,, ab ab XF in 0 C such that 12 012 ,=, ab ab Fxx xx . Thus we get a measurement 00 MO, ab CS , which clearly satisfies the (L). [Step: 3]. For each , let =1,2k 12 =1 kkk a and 1 12 = kkk b be complex numbers such that == kk ab1 . Further, consider the parallel measure- ment ,=1,2ij 0 22 MO :=,,, ij j i ab ab XXFS in the tensor -algebra * C,=1,2ij , by which the meas- ured value 8 X is obtained as follows: 11 1112 1221212222 12121212 2 ,=1,2 =,,,,,,, ij xxxxxxxxx X Here, the (7) says that, for each , ,=1,2ij 12 120 12 , 11 22 ,= , = ij ij ij ijijijij ij ab xx XX ij ij PabxxFx x Thus, putting 11 22 11 11 =1,= ,=1,= 22 abab , we calculate that 1112212 2 ,,, ,=PabPabPa bPa b 22(8) Therefore, we can conclude that (M) Bell’s inequality (i.e., a certain inequality such as “the left-hand side of (8)”, (cf. [10,11])) is broken in classical systems as well as in quantum systems. 2 s This may be somewhat significant since it says that the non-commutativity of 2 B is not necessarily indis- pensable for the (8). Although the above discussion in the steps [1-3] is easy and similar to that of [11], it should be noted that we can not assert the (M) if we do not have measurement theory (particularly, classical measurement theory ) with the interpretation (F). C opyright © 2011 SciRes. JQIS  S. ISHIKAWA Copyright © 2011 SciRes. JQIS 41 5. Conclusions Since we advanced from quantum mechanics (i.e., the mathematical formulation of Heisenberg’s uncertainty principle; [12]) to classical measurement theory, at first we had no way except relying on so-called Copenhagen interpretation in our investigations. That is, we investi- gated classical measurement theory [2-6] without the clear answer to the problem: (N) What is the Copenhagen interpretation? What is “interpretation”? Or, how should Rules 1 and 2 be used? However, in this paper we assert that we can master measurement theory thanks to man’s linguistic compe- tence if we know the interpretation (F), which may be characterized as the essence extracted from various ideas in so-called Copenhagen interpretation. Although N. Bohr said, in the Bohr-Einstein debates [13,14], that the interpretation of a physical theory has to rely on an experimental practice, we consider that the reality should be abandoned if we hope that measurement theory includes classical measurements. In this sense, we agree with A. Einstein, who never accepted the Copenha- gen interpretation as physics. That is, in spite of Bohr’s realistic view, we propose the following linguistic view: (O) In the beginning was the language called meas- urement theory (with the interpretation (F)). And, for example, quantum mechanics can be fortu- nately described in this language. And moreover, almost all scientists have already mastered this language partially and informally since statistics (at least, its basic part) is characterized as one of as- pects of measurement theory (cf. [2-6]). If it is so, measurement theory may be a miraculous language, which is common in almost all fields of sci- ence. We want to consider that this miracle was origi- nally discovered by J. von Neumann in his famous book [1]. That is because we think that measurement theory (with the interpretation (F)) is the necessary consequence of his Hilbert space formulation. Although we believe that the interpretation (F) is the unique answer to the problem (D), this should be of course examined seriously. Thus, we hope that our pro- posal (F), which is the common interpretation of classi- cal and quantum systems, will be discussed from various viewpoints. 6. Acknowledgements The author wishes to acknowledge Prof. S. Koizumi in Keio University for valuable suggestions. 7. References [1] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Springer Verlag, Berlin, 1932. [2] S. Ishikawa, “A Quantum Mechanical Approach to Fuzzy Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277-306. doi:10.1016/S0165-0114(96)00114-5 [3] S. Ishikawa, “Statistics in Measurements,” Fuzzy Sets and Systems, Vol. 116, No. 2, 2000, pp. 141-154. doi:10.1016/S0165-0114(98)00280-2 [4] S. Ishikawa, “Mathematical Foundations of Measurement Theory,” Keio University Press Inc., 2006, 335 Pages. http://www.keio-up.co.jp/kup/mfomt/). [5] S. Ishikawa, “A New Formulation of Measurement The- ory,” Far East Journal of Dynamical Systems, Vol. 10, No. 1, 2008, pp. 107-117. [6] K. Kikuchi, S. Ishikawa, “Psychological tests in meas- urement theory,” Far East Journal of Theoretical Statis- tics, Vol. 32, No. 1, 2010, pp. 81-99. [7] S. Sakai, “C*-Algebras and W*-Algebras,” Ergebnisse der Mathematik und ihrer Grenzgebiete (Band 60), Springer- Verlag, Berlin, 1971. [8] E. B. Davies, “Quantum Theory of Open Systems,” Aca- demic Press, Cambridge, 1976. [9] A. Kolmogorov, “Foundations of the Theory of Probabil- ity (Translation),” Chelsea Publishing Co., New York, 1950. [10] J. S. Bell, “On the Einstein-Podolosky-Rosen Paradox,” Physics, Vol. 1, 1966, pp. 195-200. [11] F. Selleri, “Die Debatte um die Quantentheorie,” Friedr. Vieweg & Sohn Verlagsgesellscvhaft MBH, Braun- schweig, 1983. [12] S. Ishikawa, “Uncertainty Relation in Simultaneous Measurements for Arbitrary Observables,” Reports on Mathematical Physics, Vol. 9, 1991, pp. 257-273. doi:10.1016/0034-4877(91)90046-P [13] A. Einstein, B. Podolosky and N. Rosen, “Can Quan- tum-Mechanical Description of Physical Reality Be Con- sidered Complete?” Physical Review, Vol. 47, No. 10, 1935, pp. 777-780. doi:10.1103/PhysRev.47.777 [14] N. Bohr, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Re- view, Vol. 48, 1935, pp. 696-702. doi:10.1103/PhysRev.48.696  42 S. ISHIKAWA Appendix: W*-algebraic formulation The C*-algebraic formulation (mentioned in this paper) is fundamental and essential in measurement theory. However, as mentioned in Remarks 1 and 3, the W*-algebraic formulation (cf. [5]) is, from the mathe- matical point of view, more handy than the C*-algebraic formulation (just like the Lebesgue integral is more handy than the Riemann integral). Thus we think that each of two formulations has its merits and demerits. In what follows, according to [5], in which there is a part that should be corrected, we shall add the W*-algebraic formulation. Consider the pair , H , called a basic structure. Here, is a C*-algebra, and BH is a particular C*-algebra (called a W*-algebra) such that BH is the weak closure of in BH. Let * be the pre-dual Banach space, whose existence is assured (cf. [7]). For example, we see (cf. [7]) that, when =c BH, 1) “trace class”, *= =BH, *= “trace class”. Also, when , 0 =C 2) “the space of all signed measures on *= ”, 2 =, ,LBL , =, 1 *L , where is some measure on (cf. [7]). For instance, in the above 2) we must clarify the meaning of the “value” of 0 F for ,FL and 0 * . This is easily done as follows. Let be as in (2). An element p S F *p is said to be essentially continuous at 0, if there uniquely exists a complex number S such that (*) if ,= 1 converges to * 0 p S * (i.e in the sense of weak* topology of ., 0 GG G ), then converges to . And the value of 0 is defined by the . An observable ,O:= , F in is defined as follows: 1) [ -field] is a set, , the power set of X) is a -field of X, that is, “ 12 ,, F =1nn ”, “ XF ”. 2) [Countable additivity] F is a mapping from to satisfying: a): for every , F is a non-negative element in such that 0 I , b): =0F and XI , where 0 and I is the 0-ele- ment and the identity in respectively. c): for any countable decomposition of 12 ,,, n , (i.e., , , 1, 2,3,,n=n =1 = nn = ij ij ), it holds that =1 n n FF in the sense of weak* topology in . Now we can present Rule 1 in the W*-algebraic for- mulation as follows. Rule 1' [Measurement] The probability that a meas- ured value X obtained by the measurement 0 ,S MO:= ,,XF belongs to a set is given by 0F if F is essentially co ntinuo us at * 0 p S. Next, we explain Rule 2. Let 1 1, H and 1 2 2 2,BH be basic structures. A continuous linear operator 2 1,2 : (with weak* topology) 1 (with weak* topology) is called a Markov operator, if it satis- fies that 1): 1,2 2 F0 for any non-negative element 2 in 2 , 2): = 1,2 21 I, where k is the identity in k , 2=1,k. Here note that, for any observable 22 O:=, , F in 2 , the 1,2 2 ,, F is an observable in 1 . Remark 8 In addition to the above 1) and 2), it may be natural to assume that 1,2 21 and 1,22222 2 1 supsuchthat1= 1FF F . However, from the mathematical point of view, this as- sumption is not necessarily needed. Let ,T be a tree, i.e., a partial ordered set such that “13 tt and 23 tt ” implies “1 or 21 tt2 tt ”. Here, note that T is not necessarily finite. Assume the completeness of the ordered set T. That is, for any subset TT bounded from below (i.e., there exists Tt such that tt tT T TttT ), there uniquely exists an element satisfying the following conditions, 1): , 2): if T inf inf ttT , then inf T . Definition 1' [Markov relation] The family 21 2 12 12 ,, :tt tt tt T is called a Markov relation (due to the Heisenberg picture), if it satisfies the follow- ing conditions 1) and 2). 1) With each tT , a basic structure , t t t H is associated. 2) For every 2 12 ,tt T , a Markov operator 12 ,: tt 2 t 1 t is defined. And it satisfies that 12 23 ,,tt tt 13 , =tt holds for any 12 ,tt, . 2 23 ,ttT Now Rule 2 is presented as follows: Rule 2' [Causality] The causality is represented by a Markov relation 21 2 12 12 ,, :tt tt tt T . C opyright © 2011 SciRes. JQIS
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