What are the most important anomalies across all fields that do not fit within our standard physical model?

We can start with cosmology, where dark matter and dark energy are prime examples. How accurate are the measurements pointing to the above anomalies?

Dark matter has the advantage of being confirmed by a greater variety of experiments.

A question of semantics. Anomalies should be defined as observations that theoretical models do not predict. However, both dark energy and dark matter can be easily incorporated into GR and models of particle physics.

Cosmological constant is not an anomaly but a disaster!

Condensed matter perspective: We have been very succesful at predicting properties of metals using the Fermi liquid theory. But currently more and more data are coming on a variety of states of matter, such as high-Tc superconductors, non-equilibrium phenomena etc., where there are strong deviation from the toy models.

What is the best example?

Strongly correlated electronic systems. There have been attempts to describe these using AdS/ CFT. Unlike in the standard model, it seems here one must go beyond perturbation theory to get even qualitative understanding.

What about quantum foundations and quantum gravity?

The greatest anomaly is the lack of experiments :)

The greatest challenge in quantum gravity is even a construction of a consistent theory.

In quantum foundations still have the measurement problem.

Anomalies in rotational speeds of galaxies which cannot be reconciled with any dark-matter distribution.

Many apparent anomalies in cosmology.

Dualities that appear to connect different systems that within traditional views appear uncorrelated. This puzzling fact may motivate new views in quantum field theory.

Cosmological constant problem: why is the energy density of the universe so small and makes the universe so big? Does effective field theory break down?

The puzzle of information loss in black holes. May imply modification of quantum mechanics.

This question is a bit hard to understand. Dark energy and dark matter? We don't understand those.

Entanglement isn't an anomoly, but it is a new phenomenon. We know what it is but it's new in the sense that we don't have the proper mathematical language to describe it. The standard model of particle physics introduced the Grassman field to describe Fermions and the gauge field to describe interaction. But maybe these fields can both come from entanglement... maybe at the bottom, everything is qbits.

Cosmological constant problem
• Violates separation of scales – Indicating that either general relativity does not describe gravity or QFT is incorrect.

Non – perturbative physics: Few results and is viewed as an extremely difficult area.

Need to go deep and look at the foundations of the subject, instead of just blindly building on top of previous work.

Non – equilibrium physics
• Find the correct places to look
• Topological phases of matter

Well, there aren't many anomolies, right?  Physics seems to work.  On the other hand, there is dark energy and dark matter.  But we could narrow our scope of what we consider anomolies.  Perhaps by anomolies, we mean only measurable contradictions of our theories, whereas we can easily invent kludges to explain the cosmological constant or dark matter. 

But if we don't narrow the scope of the question, we face many anomolies.  Perhaps even wavefunction collapse is an “anomoly”: it doesn't fit nicely within the traditional framework of quantum theory.  (Although the quantum foundations community has many intelligent perspectives on this “anomoly” now.)  Perhaps there are also anomolies at the interfaces of theories we usually accept as well-understood.  For instance, anything we say about quantum cosmology potentially contains anomolies.

Before we consider anyting to be an anomoloy, we should check whether its only anomoous with respect to our common sense.

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