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== Fundamental propositions == == Fundamental propositions ==
===The Third Man problem=== ===The Third Man problem===
This problem was initially discussed in detail in Everett's thesis, ''The Theory of the Universal Wavefunction''. Consider a system <math>S</math> which may take one of two states, which we shall designate <math>|A \rangle </math> and <math> |B \rangle </math>, vectors in the Hilbert space <math>H_S</math>. Now, there is an observer <math>O_1</math> who wishes to make a measurement on the system. At time <math>t_1</math>, the system may be characterised as follows: This problem was initially discussed in detail in Everett's thesis, ''The Theory of the Universal Wavefunction''. Consider a system <math>S</math> which may take one of two states, which we shall designate <math>|A \rangle </math> and <math> |B \rangle </math>, ] in the ] <math>H_S</math>. Now, there is an observer <math>O_1</math> who wishes to make a measurement on the system. At time <math>t_1</math>, the system may be characterised as follows:


<math>| \psi \rangle = \alpha|A\rangle + \beta|B\rangle </math> <math>| \psi \rangle = \alpha|A\rangle + \beta|B\rangle </math>


where <math>|\alpha|^2</math> and <math>|\beta|^2</math> 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 <math>|A\rangle</math> (but this can be substituted throughout, ''mutatis mutandis'', by <math>|B\rangle</math>). So, we may represent the sequence of event in this experiment, with observer <math>O_1</math> doing the observing, as follows: where <math>|\alpha|^2</math> and <math>|\beta|^2</math> 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 ] <math>|A\rangle</math> (but this can be substituted throughout, ''mutatis mutandis'', by <math>|B\rangle</math>). So, we may represent the sequence of event in this experiment, with observer <math>O_1</math> doing the observing, as follows:


::: <math>t_1\rightarrow t_2 </math> :::: <math>t_1\rightarrow t_2 </math>
<math>\alpha |A\rangle + \beta |B\rangle \rightarrow |A\rangle </math> <math>\alpha |A\rangle + \beta |B\rangle \rightarrow |A\rangle </math>


This is observer <math>O_1</math>'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 <math>H_S \otimes H_{O_1}</math>, where <math>H_{O_1}</math> is the Hilbert space inhabited by state vectors describing <math>O_1</math>. If the initial state of <math>O_1</math> is <math>|init\rangle</math>. After the measurement, some degrees of freedom in <math>O_1</math> become correlated with the state of <math>S</math>, and this correlation can take one of two values: <math>|O_1A\rangle</math> or <math>|O_1B\rangle</math>, with obvious meanings. If we now consider the description of the measurement event by another observer, <math>O_2</math>, This is observer <math>O_1</math>'s description of the measurement event. Now, any measurement is also a physical interaction between two or more systems. Accordingly, we can consider the ] Hilbert space <math>H_S \otimes H_{O_1}</math>, where <math>H_{O_1}</math> is the Hilbert space inhabited by state vectors describing <math>O_1</math>. If the initial state of <math>O_1</math> is <math>|init\rangle</math>. After the measurement, some degrees of freedom in <math>O_1</math> become correlated with the state of <math>S</math>, and this correlation can take one of two values: <math>|O_1A\rangle</math> or <math>|O_1B\rangle</math>, with obvious meanings. If we now consider the description of the measurement event by another observer, <math>O_2</math>, who observes the combined <math>S-O</math> system. So, the following gives the description of the measurement event according to <math>O_2</math> (again assuming that the result of the experiment gives state <math>|A\rangle</math>:


== Structure of questions == == Structure of questions ==

Revision as of 07:20, 21 July 2006

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. The interpretation was first put forward by Carlo Rovelli in 1994.

Fundamental propositions

The Third Man problem

This problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction. Consider a system S {\displaystyle S} which may take one of two states, which we shall designate | A ⟩ {\displaystyle |A\rangle } and | B ⟩ {\displaystyle |B\rangle } , vectors in the Hilbert space H S {\displaystyle H_{S}} . Now, there is an observer O 1 {\displaystyle O_{1}} who wishes to make a measurement on the system. At time t 1 {\displaystyle t_{1}} , the system may be characterised as follows:

| ψ ⟩ = α | A ⟩ + β | B ⟩ {\displaystyle |\psi \rangle =\alpha |A\rangle +\beta |B\rangle }

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

t 1 → t 2 {\displaystyle t_{1}\rightarrow t_{2}}

α | A ⟩ + β | B ⟩ → | A ⟩ {\displaystyle \alpha |A\rangle +\beta |B\rangle \rightarrow |A\rangle }

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

Structure of questions

Relationship with other interpretations

References

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