Polyakov action

Key takeaways

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

Reference excerpt

In physics, the Polyakov action is an action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by L. Brink, P. Di Vecchia and P. S. Howe in 1976, and has become associated with Alexander Polyakov after he made use of it in quantizing the string in 1981. The action reads:

S = T 2 ∫ d 2 σ − h h a b g μ ν ( X ) ∂ a X μ ( σ ) ∂ b X ν ( σ ) , {\displaystyle {\mathcal {S}}={\frac {T}{2}}\int \mathrm {d} ^{2}\sigma \,{\sqrt {-h}}\,h^{ab}g_{\mu \nu }(X)\partial _{a}X^{\mu }(\sigma )\partial _{b}X^{\nu }(\sigma ),}

where T {\displaystyle T} is the string tension, g μ ν {\displaystyle g_{\mu \nu }} is the metric of the target manifold, h a b {\displaystyle h_{ab}} is the worldsheet metric, h a b {\displaystyle h^{ab}} its inverse, and h {\displaystyle h} is the determinant of h a b {\displaystyle h_{ab}} . The metric signature is chosen such that timelike directions are + and the spacelike directions are −. The spacelike worldsheet coordinate is called σ {\displaystyle \sigma } , whereas the timelike worldsheet coordinate is called τ {\displaystyle \tau } . This is also known as the nonlinear sigma model. The Polyakov action must be supplemented by the Liouville action to describe string fluctuations.

Global symmetries N.B.: Here, a symmetry is said to be local or global from the two dimensional theory (on the worldsheet) point of view. For example, Lorentz transformations, that are local symmetries of the space-time, are global symmetries of the theory on the worldsheet.

The action is invariant under spacetime translations and infinitesimal Lorentz transformations where ω μ ν = − ω ν μ {\displaystyle \omega _{\mu \nu }=-\omega _{\nu \mu }} , and b α {\displaystyle b^{\alpha }} is a constant. This forms the Poincaré symmetry of the target manifold. The invariance under (i) follows since the action S {\displaystyle {\mathcal {S}}} depends only on the first derivative of X α {\displaystyle X^{\alpha }} . The proof of the invariance under (ii) is as follows:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Polyakov action

Start with the simplest possible case. Write down what Polyakov action claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Polyakov action 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 Polyakov action 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 Polyakov action

In research: Polyakov action appears in science 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 Polyakov action 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: Polyakov action is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conformal field theory, String theory, so understanding it makes those chapters shorter.
In everyday life: Look for Polyakov action 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.

How to study Polyakov action in 20 minutes

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

Frequently asked questions

What is Polyakov action in simple terms?

In physics, the Polyakov action is an action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by L.

Why does Polyakov action matter?

Because it connects several science 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 Polyakov action?

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 Polyakov action.