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The full expanded form of the Standard Model Lagrangian. We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density. Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an ...
In field theory, a nonlocal Lagrangian is a Lagrangian, a type of functional [()] containing terms that are nonlocal in the fields (), i.e. not polynomials or functions of the fields or their derivatives evaluated at a single point in the space of dynamical parameters (e.g. space-time). Examples of such nonlocal Lagrangians might be:
The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.
A Lagrangian density L (or, simply, a Lagrangian) of order r is defined as an n-form, n = dim X, on the r-order jet manifold J r Y of Y. A Lagrangian L can be introduced as an element of the variational bicomplex of the differential graded algebra O ∗ ∞ ( Y ) of exterior forms on jet manifolds of Y → X .
The full Lagrangian (in Euclidean signature) of the Standard model minimally coupled to gravity is obtained as pure gravity over that product space. It is therefore close in spirit to Kaluza–Klein theory but without the problem of massive tower of states.
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...
The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. [2]
Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function within that space called a Lagrangian. For many systems, L = T − V , where T and V are the kinetic and potential energy of the system, respectively.