Julian Schwinger, ''Quantum Kinematics and Dynamics''
The Classical theory of measurement is built upon the conception of an interaction between the system of interest and the measuring apparatus that can be made arbitrarily small, or at least precisely compensated, so that one can speak meaningfully of an idealized measurement that disturbs no property of the system. But it is characteristic of atomic phenomena that the interaction between system and instrument is not arbitrarily small. Nor can the disturbance produced by the interaction be compensated precisely since to some extent it is uncontrollable and unpredictable. Accordingly, a measurement on one property can produce unavoidable changes in the value previously assigned to another property, and it is without meaning to speak of a microscopic system possessing precise values for all its attributes. This contradicts the classical representation of all physical quantities by numbers. The laws of atomic physics must be expressed, therefore, in a non-classical mathematical language that constitutes a symbolic expression of the properties of microscopic measurement.
1.1 Measurement Symbols
We shall develop the outlines of this mathematical structure by discussing simplified physical systems which are such that any physical quantity A assumes only a finite number of distinct values, a', ... a^n. In the most elementary tpe of measurement, an ensemble of independent similar systems is sorted by the apparatus into subensembles, distinguished by definite values of the physical quantity being measured. Let M(a') symbolize the selective measurement that accepts systems possessing the value a' of property A and rejects all others. We define the addition of such symbols to signify less specific selective measurements that produce a subensemble associated with any of the values in the summation, none of these being distinguished by the measurement.
The multiplication of the measurement symbols represents the successive performance of measurements (read from right to left). It follows from the physical meaning of these operations that addition is commutative and associative, while multiplication is associative. With 1 and 0 symbolizing the measurements that, respectively, accept and reject all systems, the properties of the elementary selective measurements are expressed by
\begin{array}{rcl}<br />
M(a')M(a') &=& M(a')\\<br />
M(a')M(a'') &=& 0, \;\;\;a' \neq a''\\<br />
\sum M(a') &=& 1.<br />
\end{array}<br />
Indeed, the measurement symbolized by M(a') accepts every system produced by M(a') and rejects every system produced by M(a''), a'' /= a', while a selective measurement that does not distinguish any of the possible values of a' is the measurement that accepts all systems.
According to the significance of the measurements denoted as 1 and 0, these symbols have the algebraic properties:
\begin{array}{rcl}<br />
1 1 &=& 1,\\<br />
0 0 &=& 0,\\<br />
1 0 &=& 0,\\<br />
0 1 &=& 0,\\<br />
1+0 &=& 1,<br />
\end{array}
and
\begin{array}{rcl}<br />
1M(a') &=& M(a') 1 = M(a'),\\<br />
0M(a') &=& M(a') 0 = 0,\\<br />
M(a')+0 &=& M(a'),<br />
\end{array}
which justifies the notation. The various properties of 0, M(a') and 1 are consistent, provided multiplication is distributive. Thus,
\begin{array}{rcl}<br />
\sum_{a''} M(a')M(a'') &=& M(a') = M(a') 1\\<br />
&=&M(a')\sum_{a''} M(a'').<br />
\end{array}
The introduction of the numbers 1 and 0 as multipliers, with evident definitions, permits the multiplication laws of measurement symbols to be combined in the single statement
M(a')M(a'') = \delta(a',a'') M(a'),
where
\delta(a',a'') = \left[ \begin{array}{rcl}<br />
1 &,& a' = a''\\<br />
0 &,& a' \neq a''\end{array}\right.
1.2 Compatible properties. Definition of State
Two physical properties A_1 and A_2 are said to be compatible when the measurement of one does not destroy the knowledge gained by prior measurement of the other.
etc.
1.3 Measurements that Change the State
A more general type of measurement incorporates a disturbance that produces a change of state. The symbol M(a',a'') indicates a selective measurement in which systems are accepted only in the state a'' and emerge in the state a'. The measurement process M(a') is the special case for which no change of state occurs,
M(a') = M(a',a')
The properties of successive measurements of the type M(a',a'') are symbolized by
M(a',a'')M(a''',a'''') = \delta(a'',a''') M(a',a''''),\;\;\;\;\;\;\;(1.12)
for, if a'' != a''', the second stage of the compound apparatus accepts none of the systems that emerge from the first stage, while if a''=a''', all such systems enter the second stage and the compound measurement serves to select systems in the state a'''' and produce them in the state a'. Note that if the two states are reversed, we have
M(a''',a'''')M(a',a'') = \delta(a',a'''') M(a''',a''),
which differs in general from (1.12). Hence the multiplication of measurement symbols is noncommutative.
The physical quantities contained in one complete set A do not comprise the totality of physical attributes of the system. One can form other complete sets, B, C, ..., which are mutually incompatible, and for each choice of non-interfering physical characteristics there is a set of selective measurements referring to systems in the appropriate states, M(b',b''), M(c',c''), ... . The most general selective measurement involves two incompatible sets of properties. We symbolize by M(a',b') the measurement that rejects all impinging systems except those in the state b', and permits only systems in the state a' to emerge from the capparatus. The compound measurement M(a',b')M(c',d') serves to select systems in the state d' and produce them in the state a', which is a selective measurement of the type M(a',d'). But, in addition, the first stage supplies systems in the state c' while the second stage accepts only systems in the state b'. The examples of compound measurements that we have already considered involve the passage of all systems or no systems between the two stages, as represented by the multiplication of the numbers 1 and 0. More generally, measurements of properties B, performed on a system in a state c' that refers to properties incompatible with B, will yield a statistical distribution of the possible values. Hence, only a determinate fraction of the systems emerging from the first stage will be accepted by the second stage. We express this by the general multiplication law
M(a',b')M(c',d') = \langle b'|c'\rangle M(a',d'),
where \langle b'|c'\rangle is a number characterizing the statistical relation between the states b' and c'. In particular,
<br />
\langle a'|a''\rangle = \delta(a',a'').<br />
1.4 Transformation Functions
... measurement symbols of one type can be expressed as linear combinations of measurement symbols of another type. ...
1.5 The Trace
The number <a'|b'> can be regarded as a linear numerical function of the operator M(b',a'). We call this linear corresondence between operators and numbers the trace, ...
1.6 Statistical Interpretation
1.7 The Adjoint
1.8 Complex Conjugate Algebra
1.9 Matrices
1.10 Variations of Transformation Functions
1.11 Expectation Value
Chapter 2 The Geometry of States
2.1 The Null State
The uncontrollable disturbance attendant upon a measurement implies that the act of measurement is indivisible. That is to say, any attempt to trace the history of a system during a measurement process usually changes the nature of the measurement that is being performed. Hence, to conceive of a given selective measurement M(a', b') as a compound measurement is without physical implication. It is only of significance that the first stage selects systems in the state b', and that the last one produces them in the state a'; the interposed states are without meaning for the measurement as a whole. Indeed, we can even invent a non-physical state to serve as the intermediar. We shall call this mental construct the null state 0, and write
M(a',b') = M(a',0) M(0,b')
The measurement process that selects a system in the state b' and produces it in the null state,
M(0,b') = \psi(b')
can be described as the annihilation of a system in the state b'; and the production of a system in the state a' following its selection from the null state,
M(a',0) = \psi^\dag(a'),\;\;\;\;\;\;\;\;\;(2.1)
can be characterized as the creation of a system in the state a'. Thus the content of (2.1) is the indiscernability of M(a',b') from the compound process of the annihilation of a system in the state b' followed by the creation of a system in the state a',
M(a'b') = \psi(a')\psi^\dag(b').
Carl
Second, as far as I remember, background independance also is a fishy concept. Both theories will take a blow: (a) quantum mechanics needs to be undressed from it's operational formulation, meaning that you need a decent *local* theory of single events (b) GR will have to be build up from the vacuum.
