Heat Kernel & Propagator

In summary: I hope that helps.In summary, the conversation discusses the relation between the propagator of a scalar field and the heat kernel. The question is whether there is a way to invert the expression for the propagator in order to determine the corresponding heat kernel. The conversation also suggests that constructing the heat kernel would involve finding the inverse of the operator D.
  • #1
Orbb
82
0
Hi,

I have a question about the relation between the propagator of a scalar field and the heat kernel. I'm not sure wether I should rather put this question into the math section: Given a Laplacian D on some manifold M, what I mean by heat kernel is just

[tex] K(x,y;s) = \langle x | \exp(-sD) | y \rangle [/tex]

where x, y are distinct points on M and s is the diffusion time (in the sense that K obeys the heat eqn.). Now the propagator of a scalar field can be determined from K via

[tex] D^{-1}(x,y) = \int_0^{\infty} ds K(x,y;s). [/tex]

What I want to ask now is wether there is a way to invert this expression such that given some propagator, I can determine the corresponding heat kernel. Can anyobdy help?

Cheers,
O
 
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  • #2
Really no one with a hint? Is the question somehow ill-posed? Or should I add more detail about the specific calculation I am attempting?
 
  • #3
Just a guess here:
The propagator is the inverse of the operator D. So if you were given the propagator, and invert it, you would get back the operator. From there, you should be able to construct your heat kernel.
 

1. What is the Heat Kernel & Propagator?

The Heat Kernel & Propagator is a mathematical concept used in theoretical physics and mathematics to describe the behavior of heat or other physical quantities as they diffuse through space over time.

2. How is the Heat Kernel & Propagator used?

The Heat Kernel & Propagator has various applications in fields such as quantum mechanics, statistical mechanics, and differential geometry. It is used to solve differential equations and understand the behavior of heat and other physical quantities in different systems.

3. What is the difference between the Heat Kernel & Propagator?

The Heat Kernel describes the distribution of heat or other physical quantities at a specific time, while the Propagator describes how the distribution changes over time. Essentially, the Heat Kernel is a snapshot of the system at a specific time, while the Propagator shows how the system evolves over time.

4. How is the Heat Kernel & Propagator calculated?

The calculation of the Heat Kernel & Propagator involves complex mathematical equations and integrals. In simple terms, it involves solving the diffusion equation for a given system, which describes how heat or other physical quantities diffuse through space over time.

5. What are some real-world applications of the Heat Kernel & Propagator?

The Heat Kernel & Propagator have various applications in different fields such as physics, engineering, and computer science. They are used to understand the behavior of heat in materials, predict the spread of diseases, and simulate the flow of fluids. They are also used in image processing and machine learning algorithms.

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