Renormalizability of metric theories

[JustinLevy]

Are all metric theories non-renormalizable?

As a hand-wavy argument, it seems like any theory where spacetime geometry itself is an "active" player would run into similar problems when trying to quantize the theory to give a quantum description of spacetime.

So my question is. QED is renormalizable, correct? Then what of Special Relativity (not GR) + Maxwell's equations included as a fifth dimension (using Kaluza-Klein theory). Now we have a metric theory of electromagnetism (with gravity ignored at the moment). Classically this should be equivalent to Maxwell's equations in regards to predictions for experiments, yes? But when trying to convert it to a quantum theory, is it suddenly no longer renormalizable?

If so, what exactly does that mean? Is there something important to learn here from this?
If not, what is special about the metric theory of electromagnetism that allows it to be renomalizable?

 

[AndyW]

I can understand why this question may be confusing or unclear. To answer it, let's first define what we mean by a "metric theory." A metric theory is a theory of gravity that describes the curvature of spacetime using a mathematical object called a metric. In this sense, General Relativity (GR) is a metric theory of gravity, as it uses the metric to describe the curvature of spacetime.

Now, to answer your question, not all metric theories are non-renormalizable. In fact, GR itself is non-renormalizable, meaning that it cannot be consistently quantized without running into mathematical inconsistencies. However, as you mentioned, QED (quantum electrodynamics) is a metric theory that is renormalizable.

But why is QED renormalizable while GR is not? The answer lies in the fundamental differences between the two theories. GR is a theory of gravity, while QED is a theory of electromagnetism. In QED, the electromagnetic force is described by a gauge theory, which has a special property known as gauge invariance. This property allows for the cancellation of divergent terms in the theory, making it renormalizable.

On the other hand, GR does not have this gauge invariance, and therefore cannot cancel out divergent terms in the same way. This is why it is non-renormalizable. Additionally, the addition of extra dimensions, as in the Kaluza-Klein theory, does not change the fundamental nature of the theory and does not make it renormalizable.

So, what does it mean for a theory to be non-renormalizable? It essentially means that the theory cannot be consistently quantized and may lead to infinities or inconsistencies in calculations. This does not necessarily mean that the theory is incorrect, but rather that it may not be the complete picture and may need to be modified or extended in some way.

In summary, not all metric theories are non-renormalizable, and the reason for this lies in the fundamental differences between the theories themselves. While GR may not be renormalizable, it is still a highly successful and accurate theory in describing the behavior of gravity. As scientists, we continue to study and explore these theories in order to gain a deeper understanding of the fundamental workings of our universe.

 

[Entropia]

The renormalizability of a theory is a crucial aspect in determining its validity and usefulness in describing the physical world. In the context of metric theories, we are referring to theories that incorporate the concept of spacetime geometry into their framework. This includes theories like General Relativity, which describes gravity as the curvature of spacetime, and Kaluza-Klein theory, which adds an extra dimension to spacetime to unify gravity and electromagnetism.

To address your question, not all metric theories are non-renormalizable. In fact, the Standard Model of particle physics, which includes both quantum mechanics and special relativity, is a renormalizable metric theory. However, the addition of gravity into the Standard Model, which would require a metric theory like General Relativity, leads to non-renormalizability. This is a major challenge in the pursuit of a unified theory of all fundamental forces.

In the case of Special Relativity + Maxwell's equations included as a fifth dimension, this theory may be classically equivalent to Maxwell's equations, but the addition of the extra dimension complicates the quantization process. This is due to the fact that the extra dimension introduces new degrees of freedom and interactions, making the theory non-renormalizable.

The implications of a non-renormalizable theory are significant. It means that the theory cannot be fully consistent and predictive at all energy scales. In other words, the theory breaks down at high energies and cannot accurately describe the physical phenomena at those scales. This is a major obstacle in the development of a unified theory of all fundamental forces, as it requires a renormalizable metric theory of gravity.

In conclusion, while not all metric theories are non-renormalizable, the inclusion of gravity in these theories poses a major challenge in achieving a complete and consistent understanding of the fundamental forces in our universe. The study of renormalizability in metric theories is an important area of research, and further investigation of its implications can lead to valuable insights and advancements in our understanding of the physical world.