The notion of partial and complete observables, dependent and independent, is defined in Article Partial Observables of Carlo Rovelli, http://arxiv.org/abs/gr-qc/0110035 . To understand better what is an example mentioned in the article itself. Imagine a set of cards. Each has written a natural number N on one side and a natural number n on the other side. Read a letter after the other and see that there is always a correlation between N and n, ie N is a function of n, N = N (n). For example, if n less than 5, then N = 0, if n is greater than or equal to 5, then N = 1.
As defined above, n N, separately, are partial observables. We can read but can not predict them separately. However, N (n) is a completely observable. Pairs (N, n) can read but also predict, in the sense that we know that if n is a particular value, such as 7, then N will be worth N (7) = 1. If the correlation between partial observables ny N can be expressed as a function of N for n, but not of n on N, as is the case in the example, then n is an observable variable, while N is an observable dependent.
In quantum field theory partial observables are the positions and times, and the values \u200b\u200bof the field. However, there is a difference between the two, since while the field values \u200b\u200bare dependent observable, the positions and times are independent observables. Finally the value of field position and a given time is a completely observable. The important thing to note is that this refers to the quantum field theory that, somehow, represents our intuition what is absolute space and time and static fields on which they operate. However, in general relativity there are no independent observers. This is because space and time, and ultimately the gravitational field should not have a conceptual preference over other fields. Undoubtedly
the gravitational field is something special because of its universality. It is precisely the universality of the gravitational field, its effect on the entire physical body, which leads us to identify space and time as independent partial observables in quantum field theory: positions and times are somewhat independently identifiable as space-time is considered a scenario where the other interactions take place. However, in general relativity is not universality makes it a default setting but their existence and their properties are related to the existence and properties of matter, as shown in the famous hole argument of Albert Einstein.
A formal way of talking about the hole argument is dealing with the concept of diffeomorphisms. Diffeomorphisms are active coordinate transformations, while the mere coordinate transformations are passive. The mere coordinate transformation leaves the space-time coordinates moves intact, while the diffeomorphism moves all the fields of space-time coordinates and leaves intact. While a coordinate transformation can never lead to dynamic symmetry diffeomorphisms action they can and that's the main difference. The idea of \u200b\u200bdiffeomorphisms captures well the phase of the stage whatsoever, any field is relative to other points and the notion of space-time has no absolute value. Strictly
the notion of diffeomorphism
The first term represents the transformation of the variety along the integral curves of vector field:
This means that all dynamic variables in space-time are moved to along the arrows acquiring a new value. The second term of the expression of the Lie derivative is a coordinate transformation of the new values \u200b\u200bgiven by the vector field that defines the transformation to the old values. Therefore, an action is invariant under diffeomorphisms if and only if the Lie derivative of all dynamic variables or all tensor fields appearing in it is zero. And this is precisely what happens in the Einstein-Hilbert action of general relativity, and in general any action written in covariant form.
mathematical
Details: Transformations Symmetry , the Einstein-Hilbert Action, and Gauge Invariance , E. Bertschinger.
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