Role of Space Time in Jainas Syadavada and Quantum Theory

Author(s): Filita Bharucha
Publisher: Shri Satguru Publications

Page 31

Role of Quantum Theory in Deviant Logic
In studying conjunction we must distinguish between cases: (1) when the two operators commute i.e. the corresponding observables are compatible and (2) when the two operators do not commute and the corresponding observables are incompatible.
Case (1) Let the two commuting operators be denoted by A and B, they satisfy (A, B) = 0. We know from a well-known theorem in quantum mechanics that A and B have a common complete set of eigenvectors, hence we put la,> = |b> = |a, bj>l, where we have indicated the common set of eigenvectors as a, b,> in which the letters a, and b, with the same subscript 'i' are inserted with the ket symbo! 1> signifying that the ket (a, b,> is an eigenket of both A and B. Since A and B are observables, we have
1x> = ] [a,b><a,b;> \x>
A simultaneous measurement of A and B on the system S in the state 1 x> will furnish some ai as the reading of A-guage and some b; of the B-guage simultaneously, where A la, b,>= a 1 a, b> and B 1 a, b> = b, la, b> (reduction of the wave packet). In the logical language, this is a, A B. We note that the index-set 1, such that ie I, is so arranged that the results of the above measurements are a, b,. This means that when A-guage reads a,, B-guage reads b. It never happens that the results are a, and b, with i = j, hence the statement a, Ab; (i = j) is always false in this case. To construct the truth-table for conjunction ‘A’in the case of compatible observables, we consider the experimental set-up with two coincidence guages to read the values of A and B. If the A-guage reads a, and Bguage simultaneously reads b, then we assign the truth value T to both a, and B, and a, A B, is also assigned the truth value T. If the A-guage registered reading a,, but B-guage does not