{"id":1337,"date":"2026-07-20T09:14:13","date_gmt":"2026-07-20T08:14:13","guid":{"rendered":"https:\/\/vitormsalmeida.pt\/?p=1337"},"modified":"2026-07-20T09:14:13","modified_gmt":"2026-07-20T08:14:13","slug":"the-main-equations-of-string-theory","status":"publish","type":"post","link":"https:\/\/vitormsalmeida.pt\/index.php\/2026\/07\/20\/the-main-equations-of-string-theory\/","title":{"rendered":"The main equations of String Theory."},"content":{"rendered":"\n

The main equations of String Theory and their implications (an accessible explanation, but with essential mathematics).
String Theory replaces point particles with one-dimensional objects (strings) that vibrate. It is a candidate for a Theory of Everything, unifying quantum gravity with other forces. There is no single “fundamental equation” as in General Relativity (Einstein’s equations), but rather actions (functionals) that generate the equations of motion.

1. Main (fundamental) actions: <\/mark>
Nambu-Goto action (the most intuitive, area swept by the string):
S<\/mi>=<\/mo>\u2212<\/mo>T<\/mi>\u222b<\/mo>d<\/mi>2<\/mn>\u03c3<\/mi>\u2212<\/mo>d<\/mi>e<\/mi>t<\/mi>\u2061<\/mtext>(<\/mo>G<\/mi>\u03b1<\/mi>\u03b2<\/mi>)<\/mo><\/mrow>S=\u2212T\u222bd2\u03c3\u2212det\u2061(G\u03b1\u03b2)<\/annotation><\/semantics><\/math>
Where:
– (T) is the string tension (related to the parameter \u03b1\u2032\\alpha’\\alpha’ , string scale \u223c10\u221235 m).
– G<\/mi>\u03b1<\/mi>\u03b2<\/mi>=<\/mo>g<\/mi>\u03bc<\/mi>\u03bd<\/mi>(<\/mo>X<\/mi>)<\/mo>\u2202<\/mi>\u03b1<\/mi>X<\/mi>\u03bc<\/mi>\u2202<\/mi>\u03b2<\/mi>X<\/mi>\u03bd<\/mi><\/mrow>G\u03b1\u03b2=g\u03bc\u03bd(X)\u2202\u03b1X\u03bc\u2202\u03b2X\u03bd<\/annotation><\/semantics><\/math> is the induced metric in the world of the sheet (worldsheet) of the string in target spacetime with metric g<\/mi>\u03bc<\/mi>\u03bd<\/mi>g<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub>g<\/mi>\u03bc<\/mi>\u03bd<\/mi><\/mrow><\/msub><\/mrow>g\u03bc\u03bdg_{\\mu\\nu}g_{\\mu\\nu}<\/annotation><\/semantics><\/math>
– Integration over the sheet parameters (<\/mo>\u03c3<\/mi>,<\/mo>\u03c4<\/mi>\u03c3<\/mi>,<\/mo>\u03c4<\/mi>\u03c3<\/mi>,<\/mo>\u03c4<\/mi>)<\/mo><\/mrow> (\u03c3,\u03c4\\sigma, \\tau\\sigma, \\tau )<\/annotation><\/semantics><\/math>.
This action is proportional to the area swept by the string in spacetime (analogous to the worldline of a particle). It is difficult to quantize. Polyakov action (most used, classically equivalent):
S<\/mi>=<\/mo>\u2212<\/mo>T<\/mi>2<\/mn>\u222b<\/mo>d<\/mi>2<\/mn>\u03c3<\/mi>\u2212<\/mo>h<\/mi>\u2009<\/mtext>h<\/mi>a<\/mi>b<\/mi>g<\/mi>\u03bc<\/mi>\u03bd<\/mi>(<\/mo>X<\/mi>)<\/mo>\u2202<\/mi>a<\/mi>X<\/mi>\u03bc<\/mi>\u2202<\/mi>b<\/mi>X<\/mi>\u03bd<\/mi><\/mrow>S=\u2212T2\u222bd2\u03c3\u2212h\u2009habg\u03bc\u03bd(X)\u2202aX\u03bc\u2202bX\u03bd<\/annotation><\/semantics><\/math>
– h<\/mi>a<\/mi>b<\/mi><\/mrow><\/msub>h_{ab}<\/annotation><\/semantics><\/math> : auxiliary metric in the two-dimensional sheet (worldsheet).
This action is quadratically easy to quantize and has Weyl symmetries and diffeomorphisms.
Classically equivalent to the Nambu-Goto action when solving the equations for h<\/mi>a<\/mi>b<\/mi><\/mrow><\/msub>h_{ab}<\/annotation><\/semantics><\/math>.

2. Equations of motion (classical) <\/mark>
Variation of action leads to wave equations for the coordinates X\u03bc(\u03c3,\u03c4)X^\\mu(\\sigma, \\tau)X^\\mu(\\sigma, \\tau) of the string:
In the conformal gauge (simplified):
(<\/mo>\u2202<\/mi>\u03c4<\/mi>2<\/mn>\u2212<\/mo>\u2202<\/mi>\u03c3<\/mi>2<\/mn>)<\/mo>X<\/mi>\u03bc<\/mi>=<\/mo>0<\/mn><\/mrow>(\u2202\u03c42\u2212\u2202\u03c32)X\u03bc=0<\/annotation><\/semantics><\/math>(wave equation)}
With Virasoro’s restrictions (of the energy-momentum tensor T<\/mi>a<\/mi>b<\/mi>=<\/mo>0<\/mn><\/mrow> Tab=0<\/annotation><\/semantics><\/math> ):
\u2202<\/mi>+<\/mo>X<\/mi>\u22c5<\/mo>\u2202<\/mi>+<\/mo>X<\/mi>=<\/mo>0<\/mn>,<\/mo>\u2202<\/mi>\u2212<\/mo>X<\/mi>\u22c5<\/mo>\u2202<\/mi>\u2212<\/mo>X<\/mi>=<\/mo>0<\/mn><\/mrow>\u2202+X\u22c5\u2202+X=0,\u2202\u2212X\u22c5\u2202\u2212X=0<\/annotation><\/semantics><\/math>
(where \u2202<\/mi>\u00b1<\/mo>=<\/mo>\u2202<\/mi>\u03c4<\/mi>\u00b1<\/mo>\u2202<\/mi>\u03c3<\/mi><\/mrow>\u2202\u00b1=\u2202\u03c4\u00b1\u2202\u03c3<\/annotation><\/semantics><\/math> ).
In quantization, Virasoro operators and the mass spectrum (vibrational modes determine particles) emerge.

3. Crucial Quantum Consistency Conditions: <\/mark>
– Critical Dimension: To eliminate conformal anomalies (beta functions of the sigma model), bosonic theory requires 26 dimensions; superstrings (with supersymmetry) require 10 dimensions. Extra dimensions are compacted (e.g., in Calabi-Yau manifolds).
– Elimination of beta functions (\u03b2=0) of the two-dimensional sigma model: guarantees conformal invariance in the sheet. This generates Einstein’s field equations (and more) at low energy.

4. Effective low-energy action (most important implication) <\/mark>
By requiring\u03b2<\/mi>\u03bd<\/mi>\u03bc<\/mi>=<\/mo>0<\/mn><\/mrow> \u03b2\u03bd\u03bc=0<\/annotation><\/semantics><\/math> (anomaly cancellation), we obtain (in first order in \u03b1\u2032\\alpha’\\alpha’ ) the effective action in spacetime (for graviton, dilaton \u03a6, B field etc.):
S<\/mi>\u2248<\/mo>\u222b<\/mo>d<\/mi>10<\/mn>x<\/mi>\u2212<\/mo>G<\/mi>\u2009<\/mtext>e<\/mi>\u2212<\/mo>2<\/mn>\u03a6<\/mi><\/mrow>(<\/mo>R<\/mi>+<\/mo>4<\/mn>\u2202<\/mi>\u03a6<\/mi><\/mrow>\u22c5<\/mo>\u2202<\/mi>\u03a6<\/mi><\/mrow>\u2212<\/mo>112<\/mn>H<\/mi>2<\/mn>+<\/mo>O<\/mi>(<\/mo>\u03b1<\/mi>\u2032<\/mtext>)<\/mo>)<\/mo><\/mrow>S\u2248\u222bd10x\u2212G\u2009e\u22122\u03a6(R+4\u2202\u03a6\u22c5\u2202\u03a6\u2212112H2+O(\u03b1\u2032))<\/annotation><\/semantics><\/math>
Where ( R ) is the Einstein curvature scalar. The variation of this action recovers the Einstein equations (gravity) plus higher-order corrections (in \u03b1\u2032\\alpha’\\alpha’) Gravity emerges naturally from the quantum consistency of strings!

Main implications: <\/mark>
– Unification: Includes quantum gravity (graviton arises as a vibration mode), gauge forces (of open strings) and matter. Candidate for Theory of Everything.
– Supersymmetry (in superstrings): Relates bosons and fermions; aids in stability and cancellation of divergences.
– Dualities (T, S, U): Different string theories are equivalent (e.g., small ~ large strings). They lead to M-Theory (11 dimensions, includes branes).
– AdS\/CFT (Maldacena): Holographic correspondence \u2014 string theory\/gravity in Anti-de Sitter is equivalent to a conformal field theory (CFT) without gravity in the limit. Revolutionized the physics of black holes, quantum plasmas, and condensed matter.
– Landscape of universes: Thousands of ways to compactify extra dimensions \u2192 “landscape” with ~10^500 possible vacuums. Complicates predictions., but it allows for anthropia\/multiverse.
– Finite: Solves infinities of perturbative quantum gravity.
– Testability: String scale is Planckian (very small). Indirect predictions: supersymmetry (not yet observed at the LHC), extra particles, cosmological modifications, etc. Still no direct evidence.

The theory is mathematically rich (involves conformal field theory, Kac-Moody algebras, algebraic geometry), but remains speculative\u2014there is no direct experimental confirmation. Recent advances (such as bootstrap) reinforce its unique mathematical structure.

I recommend books like “String Theory” by Polchinski or Brian Greene’s explanations in “The Elegant Universe” for beginners.<\/p>\n","protected":false},"excerpt":{"rendered":"

The main equations of String Theory and their implications (an accessible explanation, but with essential mathematics). String Theory replaces point particles with one-dimensional objects (strings) that vibrate. It is a candidate for a Theory of Everything, unifying quantum gravity with other forces. There is no single “fundamental equation” as in General Relativity (Einstein’s equations), but […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[],"class_list":["post-1337","post","type-post","status-publish","format-standard","hentry","category-my-thoughts"],"_links":{"self":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts\/1337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/comments?post=1337"}],"version-history":[{"count":1,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts\/1337\/revisions"}],"predecessor-version":[{"id":1338,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/posts\/1337\/revisions\/1338"}],"wp:attachment":[{"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/media?parent=1337"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/categories?post=1337"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vitormsalmeida.pt\/index.php\/wp-json\/wp\/v2\/tags?post=1337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}