Relational Quantum Mechanics - The Problem of The Observer Observed

The Problem of The Observer Observed

This problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction. Consider observer, measuring the state of the quantum system . We assume that has complete information on the system, and that can write down the wavefunction describing it. At the same time, there is another observer, who is interested in the state of the entire - system, and likewise has complete information.

To analyse this system formally, we consider a system which may take one of two states, which we shall designate and, ket vectors in the Hilbert space . Now, the observer wishes to make a measurement on the system. At time, this observer may characterize the system as follows:

where and are probabilities of finding the system in the respective states, and obviously add up to 1. For our purposes here, we can assume that in a single experiment, the outcome is the eigenstate (but this can be substituted throughout, mutatis mutandis, by ). So, we may represent the sequence of events in this experiment, with observer doing the observing, as follows:

$\begin{matrix} t_1 & \rightarrow & t_2 \\ \alpha |\uparrow\rangle + \beta |\downarrow\rangle & \rightarrow & |\uparrow\rangle. \end{matrix}$

This is observer 's description of the measurement event. Now, any measurement is also a physical interaction between two or more systems. Accordingly, we can consider the tensor product Hilbert space, where is the Hilbert space inhabited by state vectors describing . If the initial state of is, some degrees of freedom in become correlated with the state of after the measurement, and this correlation can take one of two values: or where the direction of the arrows in the subscripts corresponds to the outcome of the measurement that has made on . If we now consider the description of the measurement event by the other observer, who describes the combined system, but does not interact with it, the following gives the description of the measurement event according to, from the linearity inherent in the quantum formalism:

$\begin{matrix} t_1 & \rightarrow & t_2 \\ \left( \alpha | \uparrow \rangle + \beta | \downarrow \rangle \right) \otimes | init \rangle & \rightarrow & \alpha | \uparrow \rangle \otimes | O_{\uparrow} \rangle + \beta | \downarrow \rangle \otimes | O_{\downarrow} \rangle. \end{matrix}$

Thus, on the assumption (see hypothesis 2 below) that quantum mechanics is complete, the two observers and give different but equally correct accounts of the events .

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