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Relational quantum mechanics is an interpretation of quantum mechanics which is distinguished by its construal of the state of a quantum system as being the correlation between this system and an observer.

Overview of the Theory

Derivation & Structure

The Third Man problem

This problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction. Consider a system ${\displaystyle S}$ which may take one of two states, which we shall designate ${\displaystyle |A\rangle }$ and ${\displaystyle |B\rangle }$, vectors in the Hilbert space ${\displaystyle H_{S}}$. Now, there is an observer ${\displaystyle O_{1}}$ who wishes to make a measurement on the system. At time ${\displaystyle t_{1}}$, the system may be characterised as follows:

${\displaystyle |\psi \rangle =\alpha |A\rangle +\beta |B\rangle }$

where ${\displaystyle |\alpha |^{2}}$ and ${\displaystyle |\beta |^{2}}$ 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 ${\displaystyle |A\rangle }$ (but this can be substituted throughout, mutatis mutandis, by ${\displaystyle |B\rangle }$). So, we may represent the sequence of event in this experiment, with observer ${\displaystyle O_{1}}$ doing the observing, as follows:

${\displaystyle t_{1}\rightarrow t_{2}}$

${\displaystyle \alpha |A\rangle +\beta |B\rangle \rightarrow |A\rangle }$

This is observer ${\displaystyle O_{1}}$'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 ${\displaystyle H_{S}\otimes H_{O_{1}}}$, where ${\displaystyle H_{O_{1}}}$ is the Hilbert space inhabited by state vectors describing ${\displaystyle O_{1}}$. If the initial state of ${\displaystyle O_{1}}$ is ${\displaystyle |init\rangle }$. After the measurement, some degrees of freedom in ${\displaystyle O_{1}}$ become correlated with the state of ${\displaystyle S}$, and this correlation can take one of two values: ${\displaystyle |O_{1}A\rangle }$ or ${\displaystyle |O_{1}B\rangle }$, with obvious meanings. If we now consider the description of the measurement event by another observer, ${\displaystyle O_{2}}$, who observes the combined ${\displaystyle S-O}$ system. So, the following gives the description of the measurement event according to ${\displaystyle O_{2}}$ (again assuming that the result of the experiment gives state ${\displaystyle |A\rangle }$:

Fundamental Propositions

Example: EPR

Relationship with other interpretations

Wave Collapse

No Wave Collapse

References

Get Everett, Rovelli (Int Jour Theor phys), von Neumann.