Solve Path Integrals: Find Schroedinger Equation & Quantum Gravity

In summary, experts have attempted to use path integrals to find the Schrodinger equation for quantum gravity. However, this approach is limited to linearized general relativity and is ultimately non-renormalizable. Attempting to get a Schrodinger equation for quantum gravity raises new problems and the development of a background-independent quantum mechanics has been a recent challenge. Abhay Ashtekar has made progress in constructing a quantum theory of gravity, but it is still a work in progress due to the complexities of gravity, even in its classical form.
  • #1
eljose79
1,518
1
If you have path integrals could you find the schroedinguer equation?..in fact why is not this made to find the equation in quantum gravity?..
 
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  • #2
To set up the integrals you need the potentials. It seems to me that you have to have some equation for the potentials before you can do the integrals. Am I right or wrong? Experts?
 
  • #3
If you treat general relativity (GR) as a field theory, trying to quantize it using path integrals, the best you can do is to describe the quantization of *linearized* GR, meaning you look at small fluctuations about a smooth background spacetime (described by a metric, either flat or curved...let's just stick with the Minkowski metric). Then if you carry out the path integral quantization, and get past all the subtleties involved, you find that the theory is ultimately non-renormalizable. By our interpretation and criterion of renormalizability for a "physical theory", we conclude that such an attempt (to describe quantum general relativity as a field theory on some smooth, classical, spacetime) is wrong.
If you try to go back and get a "Schrodinger equation" for quantum gravity, describing the evolution of the metric you reach whole new problems that are too detailed to go into here. The problem is that we didn't have a background independent quantum mechanics not too long ago. However, in the past two decades, Abhay Ashtekar has been greatly responsible for constructing such a quantum theory of gravity (though the story is far from over and it is not nearly a final model of quantum gravity). Gravitation is not a simply theory, even classical GR has thorny issues, let alone the quantum theory!
 

FAQ: Solve Path Integrals: Find Schroedinger Equation & Quantum Gravity

1. What is a path integral?

A path integral is a mathematical technique used in quantum mechanics to calculate the probability of a particle moving from one point to another in space and time. It involves summing up all possible paths that the particle could take between the two points, taking into account the probabilities associated with each path.

2. How do you solve path integrals?

To solve a path integral, you first need to determine the action of the system, which is a mathematical expression that describes the energy of the system. Then, you need to sum up all possible paths using the action as a weight, and integrate over all possible paths. This results in a probability amplitude, which can then be used to calculate the probability of the particle's motion.

3. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the evolution of a quantum system over time. It relates the time derivative of the wave function, which describes the state of the system, to the energy of the system and the potential energy function.

4. How does solving path integrals relate to quantum gravity?

Solving path integrals is an important tool in understanding the behavior of particles at the quantum level. In the context of quantum gravity, it allows us to study the behavior of particles in the presence of spacetime curvature, which is a key aspect of Einstein's theory of general relativity. By applying the principles of quantum mechanics to gravity, we can gain insights into the nature of the universe at a fundamental level.

5. What are the practical applications of solving path integrals?

Solving path integrals has a wide range of practical applications, including in the development of quantum technologies such as quantum computers and quantum communication. It also allows us to make precise predictions about the behavior of particles in various physical systems, which has implications for fields such as materials science and chemistry. Additionally, path integrals are used in the development of theories such as quantum field theory and quantum gravity, which aim to better understand the fundamental workings of our universe.

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