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http://arxiv.org/abs/1404.1750
How many quanta are there in a quantum spacetime?
Seramika Ariwahjoedi, Jusak Sali Kosasih, Carlo Rovelli, Freddy P. Zen
(Submitted on 7 Apr 2014)
Following earlier insights by Livine and Terno, we develop a technique for describing quantum states of the gravitational field in terms of coarse grained spin networks. We show that the number of nodes and links and the values of the spin depend on the observables chosen for the description of the state. Hence the question in the title of this paper is ill posed, unless further information about what is being measured is given.
16 pages, 9 figures
==quote page 13, section VII==
VII. HOW MANY QUANTA OF SPACE ARE THERE IN QUANTUM SPACETIME?
Armed with the observations of Section II and with the technology developed in Sections III to VI, let us return to the question of the number of quanta in a given classical geometry. The second example of Section II (the oscillators) shows that the number of quanta is not an absolute property of a quantum state: it depends on the basis on which the state is expanded. In turn, this depends on the way we are interacting with the system.
... For instance, an antenna tuned into a wavelength λ is insensitive to the high-frequency components of the field, if any of these is excited. We cannot treat the gravitational field in the same manner at all scales, because Fourier analysis requires a background geometry, which is, in general, not available in gravity.
Sections III to VI, however, provide a viable alternative: areas and volumes of big links and big nodes, AL and VN , capture large scale features of the field, and are insensitive to higher frequency components of the field, in a way similar to the long wavelength Fourier modes. In fact, notice that this is what we mean when we refer to macroscopic areas and volume.
The area of a table is not the sum of individual areas of all microscopic elements of its boundary; it is the area of a coarse-grained description of the table where the surface is assumed to be flat, even if in reality it is not, at small scales. When we measure the gravitational field, that is, geometrical quantities, we routinely refer to its long wavelength modes. For instance, we can measure the Earth-Moon distance with a laser. What we are measuring is a non-local, integrated value of the gravitational field, in the same manner in which an antenna measures a single wavelength of the electromagnetic field.
The quanta of the gravitational field we interact with, are those described by the quantum numbers of coarse-grained operators like AL and VN , not the maximally fine-grained ones.
Therefore we can begin to answer the question of the title. The number of quanta we see in a system depends on the way we interact with it. When interacting with a gravitational field at large scales we are probing coarse grained features of space, which can be described by the quantum numbers JL,VN of a coarse-grained graph γ0. Probing the field as shorter scales tests higher modes, which can be described by more fine grained subset graphs γ1,...,γm.
The relevance of this construction for understanding the scaling the dynamics, cosmology and black holes will be studied elsewhere.
==endquote==
How many quanta are there in a quantum spacetime?
Seramika Ariwahjoedi, Jusak Sali Kosasih, Carlo Rovelli, Freddy P. Zen
(Submitted on 7 Apr 2014)
Following earlier insights by Livine and Terno, we develop a technique for describing quantum states of the gravitational field in terms of coarse grained spin networks. We show that the number of nodes and links and the values of the spin depend on the observables chosen for the description of the state. Hence the question in the title of this paper is ill posed, unless further information about what is being measured is given.
16 pages, 9 figures
==quote page 13, section VII==
VII. HOW MANY QUANTA OF SPACE ARE THERE IN QUANTUM SPACETIME?
Armed with the observations of Section II and with the technology developed in Sections III to VI, let us return to the question of the number of quanta in a given classical geometry. The second example of Section II (the oscillators) shows that the number of quanta is not an absolute property of a quantum state: it depends on the basis on which the state is expanded. In turn, this depends on the way we are interacting with the system.
... For instance, an antenna tuned into a wavelength λ is insensitive to the high-frequency components of the field, if any of these is excited. We cannot treat the gravitational field in the same manner at all scales, because Fourier analysis requires a background geometry, which is, in general, not available in gravity.
Sections III to VI, however, provide a viable alternative: areas and volumes of big links and big nodes, AL and VN , capture large scale features of the field, and are insensitive to higher frequency components of the field, in a way similar to the long wavelength Fourier modes. In fact, notice that this is what we mean when we refer to macroscopic areas and volume.
The area of a table is not the sum of individual areas of all microscopic elements of its boundary; it is the area of a coarse-grained description of the table where the surface is assumed to be flat, even if in reality it is not, at small scales. When we measure the gravitational field, that is, geometrical quantities, we routinely refer to its long wavelength modes. For instance, we can measure the Earth-Moon distance with a laser. What we are measuring is a non-local, integrated value of the gravitational field, in the same manner in which an antenna measures a single wavelength of the electromagnetic field.
The quanta of the gravitational field we interact with, are those described by the quantum numbers of coarse-grained operators like AL and VN , not the maximally fine-grained ones.
Therefore we can begin to answer the question of the title. The number of quanta we see in a system depends on the way we interact with it. When interacting with a gravitational field at large scales we are probing coarse grained features of space, which can be described by the quantum numbers JL,VN of a coarse-grained graph γ0. Probing the field as shorter scales tests higher modes, which can be described by more fine grained subset graphs γ1,...,γm.
The relevance of this construction for understanding the scaling the dynamics, cosmology and black holes will be studied elsewhere.
==endquote==
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