physics

Ultraviolet fixed point

Ultraviolet fixed point is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ultraviolet fixed point rather than just read about it. In short: In a quantum field theory, one may calculate an effective or running coupling constant that defines the coupling of the theory measured at a given momentum scale. One example of such a coupling constant is the electric charge.

Key takeaways

Ultraviolet fixed point belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ultraviolet fixed point to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ultraviolet fixed point from memory before moving on to harder problems.

Reference excerpt

In a quantum field theory, one may calculate an effective or running coupling constant that defines the coupling of the theory measured at a given momentum scale. One example of such a coupling constant is the electric charge. In approximate calculations in several quantum field theories, notably quantum electrodynamics and theories of the Higgs particle, the running coupling appears to become infinite at a finite momentum scale. This is sometimes called the Landau pole problem. It is not known whether the appearance of these inconsistencies is an artifact of the approximation, or a real fundamental problem in the theory. However, the problem can be avoided if an ultraviolet or UV fixed point appears in the theory. A quantum field theory has a UV fixed point if its renormalization group flow approaches a fixed point in the ultraviolet (i.e. short length scale/large energy) limit. This is related to zeroes of the beta-function appearing in the Callan–Symanzik equation. The large length scale/small energy limit counterpart is the infrared fixed point.

Specific cases and details Among other things, it means that a theory possessing a UV fixed point may not be an effective field theory, because it is well-defined at arbitrarily small distance scales. At the UV fixed point itself, the theory can behave as a conformal field theory. The converse statement, that any QFT which is valid at all distance scales (i.e. isn't an effective field theory) has a UV fixed point is false. See, for example, cascading gauge theory. Noncommutative quantum field theories have a UV cutoff even though they are not effective field theories. Physicists distinguish between trivial and nontrivial fixed points. If a UV fixed point is trivial (generally known as Gaussian fixed point), the theory is said to be asymptotically free. On the other hand, a scenario, where a non-Gaussian (i.e. nontrivial) fixed point is approached in the UV limit, is referred to as asymptotic safety. Asymptotically safe theories may be well defined at all scales despite being nonrenormalizable in perturbative sense (according to the classical scaling dimensions).

Asymptotic safety scenario in quantum gravity

Steven Weinberg has proposed that the problematic UV divergences appearing in quantum theories of gravity may be cured by means of a nontrivial UV fixed point. Such an asymptotically safe theory is renormalizable in a nonperturbative sense, and due to the fixed point physical quantities are free from divergences. As yet, a general proof for the existence of the fixed point is still lacking, but there is mounting evidence for this scenario.

See also Asymptotic freedom Asymptotic safety in quantum gravity Landau pole Quantum triviality Ultraviolet divergence

References

Worked examples

Example 1 — a first encounter with Ultraviolet fixed point

Start with the simplest possible case. Write down what Ultraviolet fixed point claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ultraviolet fixed point before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ultraviolet fixed point ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ultraviolet fixed point

In research: Ultraviolet fixed point appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ultraviolet fixed point in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ultraviolet fixed point is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conformal field theory, Fixed points (mathematics), Quantum physics stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Ultraviolet fixed point outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Ultraviolet fixed point in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ultraviolet fixed point means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ultraviolet fixed point out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ultraviolet fixed point in simple terms?

Why does Ultraviolet fixed point matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ultraviolet fixed point?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ultraviolet fixed point.