mathematics

Weinberg–Witten theorem

Weinberg–Witten theorem is a mathematics 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 Weinberg–Witten theorem rather than just read about it. In short: In theoretical physics, the Weinberg–Witten (WW) theorem, proved by Steven Weinberg and Edward Witten, states that massless particles (either composite or elementary) with spin j > 1/2 cannot carry a Lorentz-covariant current, while massless particles with spin j > 1 cannot carry a Lorentz-covariant stress-energy. The theorem is usually interpreted to mean that the graviton (j = 2) cannot be a composite particle in…

Key takeaways

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

Reference excerpt

In theoretical physics, the Weinberg–Witten (WW) theorem, proved by Steven Weinberg and Edward Witten, states that massless particles (either composite or elementary) with spin j > 1/2 cannot carry a Lorentz-covariant current, while massless particles with spin j > 1 cannot carry a Lorentz-covariant stress-energy. The theorem is usually interpreted to mean that the graviton (j = 2) cannot be a composite particle in a relativistic quantum field theory.

Background During the 1980s, preon theories, technicolor and the like were very popular and some people speculated that gravity might be an emergent phenomenon or that gluons might be composite. Weinberg and Witten, on the other hand, developed a no-go theorem that excludes, under very general assumptions, the hypothetical composite and emergent theories. Decades later new theories of emergent gravity are proposed and some high-energy physicists are still using this theorem to try and refute such theories. Because most of these emergent theories aren't Lorentz covariant, the WW theorem doesn't apply. The violation of Lorentz covariance, however, usually leads to other problems.

Theorem Weinberg and Witten proved two separate results. According to them, the first is due to Sidney Coleman, who did not publish it:

A 3 + 1D QFT (quantum field theory) with a conserved 4-vector current J μ {\displaystyle J^{\mu }} (see four-current) which is Poincaré covariant (and gauge invariant if there happens to be any gauge symmetry which hasn't been gauge-fixed) does not admit massless particles with helicity |h| > 1/2 that also have nonzero charges associated with the conserved current in question. A 3 + 1D QFT with a non-zero conserved stress–energy tensor T μ ν {\displaystyle T^{\mu \nu }} which is Poincaré covariant (and gauge invariant if there happens to be any gauge symmetry which hasn't been gauge-fixed) does not admit massless particles with helicity |h| > 1.

A sketch of the proof The conserved charge Q is given by ∫ d 3 x J 0 {\displaystyle \int d^{3}x\,J^{0}} . We shall consider the matrix elements of the charge and of the current J μ {\displaystyle J^{\mu }} for one-particle asymptotic states, of equal helicity, | p ⟩ {\displaystyle |p\rangle } and | p ′ ⟩ {\displaystyle |p'\rangle } , labeled by their lightlike 4-momenta. We shall consider the case in which ( p − p ′ ) {\displaystyle (p-p')} isn't null, which means that the momentum transfer is spacelike. Let q be the eigenvalue of those states for the charge operator Q, so that:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Weinberg–Witten theorem

Start with the simplest possible case. Write down what Weinberg–Witten theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Weinberg–Witten theorem 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 Weinberg–Witten theorem 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 Weinberg–Witten theorem

In research: Weinberg–Witten theorem appears in mathematics 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 Weinberg–Witten theorem 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: Weinberg–Witten theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics No-go theorems, Quantum field theory, Quantum gravity, so understanding it makes those chapters shorter.
In everyday life: Look for Weinberg–Witten theorem 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 Weinberg–Witten theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Weinberg–Witten theorem 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 Weinberg–Witten theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Weinberg–Witten theorem in simple terms?

Why does Weinberg–Witten theorem matter?

Because it connects several mathematics 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 Weinberg–Witten theorem?

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 Weinberg–Witten theorem.