By Julian Schwinger
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Example text
A. M. Dirae, Be», Mad. Phys, 17, 19S (1945), where n<»n-Hermitian operators and complex "probabilities" are introduced. 8 The Null State Reconstruction of the Measurement Algebra Vector Algebra Wave Functions Unitary Transformations Infinitesimal Unitary Transformations Successive Unitary Transformations Unitary Transformation GroupsTranslations and Rotations 2 . 1 THE NULL STATE 29 32 35 37 40 44 46 48 53 54 56 62 The uncontrollable disturbance attendant upon a measurement implies that the act of measurement is indivisible.
This geometry is metrical since the number defines a scalar product. 27) describe a change in coordinate system. The product of an operator with a vector expresses THE GEOMETRY OF STATES 37 a mapping upon another vector in the same space, The effect on the vectors of the a-coordinate system of the operator symbolizing property A , is given by which characterizes f(a1) and $(a') as the right and left eigenvectors, respectively, of the complete set of commuting operators eigenvalues a' A , with the .
The elements of the measure- ment algebra are called operators. 5 THE TRACE The number { a'|b') can be regarded as a linear numerical function of the operator M(b'» a'). 27). In particular, The trace of a measurement symbol product is which can be compared with Hence, despite the noncommutativity of multiplication, the trace of a product of two factors is independent of the multiplication order. 6 STATISTICAL INTERPRETATION It should be observed that the general multi- plication law and the definition of the trace are preserved if we make the substitutions where the numbers X(a') and X(b') can be given arbitrary non-zero values.