Why the term cylindrical functions for the states in LQG?

In summary, the term "cylindrical functions" is used in LQG to refer to functions that only depend on finite coordinates and can be projected onto a finite dimensional subspace. This concept is inspired by the use of cylinder functions in constructing measures on infinite dimensional vector spaces.
  • #1
zwicky
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Why the term "cylindrical functions" for the states in LQG?

I just can't find a any reference that explain the why of the term "cylindrical" for the states in LQG. Thanks in advance for any comment.

Z.
 
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  • #2


zwicky said:
I just can't find a any reference that explain the why of the term "cylindrical" for the states in LQG. Thanks in advance for any comment.

Z.
The term means different things in different contexts. The term "cylinder function" comes up in constructing measures on infinite dimensional vector spaces.

For example here is a reference to 1957 Proceedings of the National Academy of Sciences
http://www.ncbi.nlm.nih.gov:80/pmc/articles/PMC528446/?page=1

Kurt Friedrichs and Howard Shapiro
Integration over Hilbert Space and Outward Extensions.

The Hilbertspace H is infinite dimensional, and you want to be able to do integral calculus on it. You want to be able to integrate functions defined on the Hilbertspace.
You start with a simple set of functions. A function f is a cylinder function based on E, if there is a finite dimensional subspace E, of H, such that if PE is the projection of H down onto E, then for all x in H we have f(x) = f(PE x).

It is a beautifully simple idea, it means that the function f depends only on finite coordinates. You can project down which means throwing away all the other information and the function only depends on x thru its projection onto the subspace E.

I have to go. But Weiner measure on random walks and all kinds of stochastic process measures on various function spaces can be constructed this way.

It is a classic idea in topological vectorspaces.

I think they just took it over. I will check in later and clarify as needed.
===EDIT==
Checked in the next day.
Well maybe the analogy with what happens with cylinder functions in Loop is obvious. I don't see more questions. So maybe this explanation is OK.
 
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  • #3


The term "cylindrical functions" in LQG (Loop Quantum Gravity) refers to the mathematical functions that are used to describe the quantum states of the gravitational field. These functions are defined on a three-dimensional cylindrical space, hence the name "cylindrical functions". This choice of mathematical framework is based on the fact that in LQG, the fundamental variables are the connection and the triad, which are closely related to the geometry of a three-dimensional cylindrical space.

Furthermore, these cylindrical functions have specific properties that make them well-suited for the quantization of the gravitational field. They are holomorphic functions, meaning they are complex-valued and analytic, and they have a specific transformation property under diffeomorphism, which is a key symmetry in general relativity. This allows for a consistent and covariant quantization of the gravitational field.

The term "cylindrical functions" may also be used to distinguish them from other types of functions used in other quantum gravity theories, such as spin networks in the spin foam approach. In LQG, the cylindrical functions are considered to be more fundamental, as they are used to construct the physical states of the gravitational field.

In summary, the term "cylindrical functions" in LQG refers to the specific mathematical functions that are used to describe the quantum states of the gravitational field, and their name is derived from their properties and their role in the mathematical framework of LQG.
 

Related to Why the term cylindrical functions for the states in LQG?

1. What are cylindrical functions in the context of LQG?

Cylindrical functions are a type of mathematical function used in the framework of loop quantum gravity (LQG) to describe the quantum states of a physical system. These functions are defined on a topological space known as a cylindrical manifold, which has the shape of a cylinder. In LQG, cylindrical functions are used to represent the quantum states of space and provide a basis for the quantum description of physical quantities such as area and volume.

2. Why are cylindrical functions used in LQG?

Cylindrical functions are used in LQG because they have certain mathematical properties that make them well-suited for describing the quantum states of a physical system. These functions are able to capture the discrete nature of space and provide a natural way to define physical quantities in a quantum framework. Additionally, cylindrical functions have been shown to have a close connection to the geometry of space, which is a central concept in LQG.

3. How are cylindrical functions defined in LQG?

In LQG, cylindrical functions are defined on a cylindrical manifold, which is a topological space that has the shape of a cylinder. These functions are characterized by their dependence on the holonomies of connections, which are mathematical objects that describe how a field varies over a path in space. The precise mathematical definition of cylindrical functions involves a combination of these holonomies with certain algebraic operations.

4. What makes cylindrical functions special in LQG?

Cylindrical functions are special in LQG because they are able to capture the discrete nature of space and provide a natural way to define physical quantities in a quantum framework. This is a unique feature of LQG, as other approaches to quantum gravity, such as string theory, do not have a similar mathematical structure. Additionally, cylindrical functions have been shown to have a deep connection to the geometry of space, which is a fundamental concept in LQG.

5. How do cylindrical functions relate to other mathematical objects in LQG?

In LQG, cylindrical functions are closely related to other mathematical objects such as spin networks and spin foam states. These objects provide a basis for the quantum description of space and are used to define physical quantities in a discrete and quantum framework. Cylindrical functions can be seen as a more general and flexible mathematical tool that allows for a wider range of physical quantities to be described in LQG.

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