Lagrangian (field theory)
In 1980, Claude Itykson and Jean-Bernard Zuber published a text that defined the modern approach to Lagrangian field theory. This formalism serves as the field-theoretic analogue of Lagrangian mechanics used for discrete particles. Discrete particle systems possess a finite number of degrees of freedom. Field theories apply to continua and fields with an infinite number of degrees of freedom. The independent variable in field theory is replaced by an event in spacetime or a point on a Riemannian manifold. Dependent variables become the value of a field at that specific point. Equations of motion are obtained through an action principle written as an integral over this domain. The calligraphic typeface denotes the density while brackets denote integration limits. A volume form measures the domain of the field function itself.
Mathematical formulations often express the Lagrangian as a function on a fiber bundle. Euler, Lagrange equations can be interpreted as specifying geodesics on this fiber bundle. Topics like tangent manifolds, symplectic manifolds, and contact geometry emerge from this geometric structure. Scalar fields are understood as coordinates on a fiber bundle. Derivatives of the field are sections of the jet bundle. In mathematical literature, spacetime is taken to be a Riemannian manifold. The integral then becomes a volume form involving the wedge product. The square root of the determinant of the metric tensor appears within this form. For flat spacetime such as Minkowski space, the unit volume is one and commonly omitted. Some older textbooks write the volume form with a minus sign appropriate for certain metric signatures. When discussing general Riemannian manifolds, the Hodge star operator provides an abbreviated notation. This geometric clarity allows abstract theorems from geometry to gain insight into physical systems.
Often the Lagrangian consists of a sum of polynomial terms dictated by symmetries. Terms containing the product of two fields and no derivatives are known as mass terms. These give mass to the fields themselves. Other terms with at least one derivative are known as kinetic terms. They make fields dynamical while most theories restrict them to at most two derivatives. Kinetic terms are usually positive-definite to ensure positive energies. Fields with no kinetic terms play roles as auxiliary or background fields. Any term with more than two fields per term is known as an interaction term. Coefficients in front of these terms are coupling constants dictating interaction strength. A quartic interaction in real scalar field theory involves a specific coupling constant value. Interacting terms can have any number of derivatives providing momentum dependence. Terms with only one field are known as tadpole terms since they produce tadpole Feynman diagrams. Constant terms with no fields have no physical consequences in non-gravitational theories. In gravitational systems, constant terms couple to spacetime via the metric determinant. They play the role of the cosmological constant affecting dynamics at both classical and quantum levels.
Consider a point particle interacting with the electromagnetic field through continuous charge density. The resulting Lagrangian density for electromagnetism includes a current density measured in amperes per square meter. Varying this action yields Gauss' law when done with respect to the potential. Varying instead with respect to the vector potential yields Ampère's law. Tensor notation packages charge density into a current four-vector and potential into a potential four-vector. These vectors form inner products within the Minkowski metric framework. The electric and magnetic fields combine into what is known as the electromagnetic tensor. This tensor defines Maxwell's equations using the Levi-Civita tensor. Classical electromagnetism becomes a Lorentz-invariant theory under these conditions. Using differential forms allows writing the electromagnetic action without an additional integration measure. Forms have coordinate differentials built directly into their structure. Variation of the action leads to Maxwell's equations for the electromagnetic potential. Substituting the field strength immediately yields the equation for the fields themselves. The A field can be understood as the affine connection on a U(1)-fiber bundle over spacetime.
The Yang, Mills equations can be written in exactly the same form as above by replacing Lie groups. In the Standard Model, it is conventionally taken to be SU(3) or similar structures. No quantization needs to be performed despite historical roots in quantum field theory. Chern, Simons functional considers actions in one dimension less within contact geometry settings. Ginzburg, Landau Lagrangian combines scalar field theory with Yang-Mills action. It may be written where a section of a vector bundle corresponds to the order parameter in a superconductor. The second term represents the famous Sombrero hat potential associated with Higgs fields. Dirac spinors appear as special cases constructed from Clifford algebra of spacetime. Weyl spinors provide a more general foundation usable in any number of dimensions. Quantum chromodynamic Lagrangian combines massive Dirac spinors with Yang-Mills action describing gauge field dynamics. The combined Lagrangian remains gauge invariant regardless of quark types counted up to six. Historical artifacts like the word quantum only acknowledge development paths rather than requiring actual quantization. Full gauge-invariant classical formulations exist for these systems without invoking particle creation or annihilation.
The Lagrange density for general relativity includes the cosmological constant and curvature scalar. This integral is known as the Einstein, Hilbert action involving Riemann tensor contractions. Christoffel symbols define the metric connection on spacetime itself. Moving bodies follow geodesics on manifolds described by connections rather than force fields. Substituting this Lagrangian into Euler, Lagrange equations yields Einstein field equations when taking the metric tensor as the field. Energy momentum tensors are defined by determinants of the metric matrix. Integration measures become coordinate independent through root factors of metric determinants. Electromagnetism in general relativity contains the Einstein, Hilbert action from above. Pure electromagnetic Lagrangians serve as matter Lagrangians within curved spacetime frameworks. Solving both Einstein and Maxwell's equations around spherically symmetric mass distributions leads to Reissner, Nordström charged black holes. Kaluza, Klein theory attempts unification using a fifth dimension to construct affine bundles. Factorizations such as 7-sphere products accounted for early excitement about theories of everything. The 7-sphere proved not large enough to enclose all Standard Model components, dashing these hopes.
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Common questions
Who published the text that defined the modern approach to Lagrangian field theory in 1980?
Claude Itykson and Jean-Bernard Zuber published a text in 1980 that defined the modern approach to Lagrangian field theory. This formalism serves as the field-theoretic analogue of Lagrangian mechanics used for discrete particles.
What is the independent variable in Lagrangian field theory compared to discrete particle systems?
The independent variable in field theory is replaced by an event in spacetime or a point on a Riemannian manifold. Field theories apply to continua and fields with an infinite number of degrees of freedom unlike discrete particle systems which possess a finite number of degrees of freedom.
How does the Lagrange density for general relativity relate to the Einstein Hilbert action?
The Lagrange density for general relativity includes the cosmological constant and curvature scalar within the integral known as the Einstein Hilbert action involving Riemann tensor contractions. Substituting this Lagrangian into Euler Lagrange equations yields Einstein field equations when taking the metric tensor as the field.
Why do terms containing the product of two fields and no derivatives give mass to fields themselves?
Terms containing the product of two fields and no derivatives are known as mass terms because they give mass to the fields themselves. Other terms with at least one derivative are known as kinetic terms that make fields dynamical while most theories restrict them to at most two derivatives.
What role does the A field play in the context of fiber bundles over spacetime?
The A field can be understood as the affine connection on a U(1)-fiber bundle over spacetime. This formulation allows writing the electromagnetic action without an additional integration measure using differential forms that have coordinate differentials built directly into their structure.