Lagrangian covariant

A covariant Lagrangian density (i.e. an equation that looks the same in all Lorentz frames) will be obtained if the Lagrangian density is defined in terms of a relativistic scalar density, as in... 

Sg covariance matrix of the error estimates covariance matrix of variable estimates X Lagrangian multipliers... 

Here,, 4 ( is the vector 4-potential introduced in the vacuum as part of the covariant derivative, and therefore introduced by spacetime curvature. The electromagnetic field and the topological charge g are the results of the invariance of the Lagrangian (868) under local U(l) gauge transformation, in other words, the results of spacetime curvature. 

The effect of the local gauge transform is to introduce an extra term 8M A in the transformation of the derivatives of fields. Therefore, 8 A does not transform covariantly, that is, does not transform in the same way as A itself. These extra terms destroy the invariance of the action under the local gauge transformation, because the change in the Lagrangian is... 

Following in analogy with the theory of weak interactions we let / be a doublet that describes an electron according to the 1 field and the 3 field. Here we illustrate the sort of physics that would occur with a chiral theory. We start with the free particle Dirac Lagrangian and let the differential become gauge covariant,... 

The key feature of the theory of QED—whether it is cast in nonrelativis-tic or fully covariant forms is that the electromagnetic field obeys quantum mechanical laws. A frequent first step in the construction of either version of the theory is the writing of the classical Lagrangian function for the interaction of a charged particle with a radiation field. For a particle of mass m, electronic charge —e, located at position vector q, and moving with velocity d /df c in a position-dependent potential V( ) subject to electromagnetic radiation described by scalar and vector potentials cp0) and a(r), at field point... 

The Lagrangian density that gives, on variation, the topologically covariant field equations (33) is an explicit function of the spinor variables, j, if), 2,4>2 and their respective covariant derivatives. (The dagger superscript denotes the Hermitian conjugate of the function.) It has the form ... 

The freedom to choose any of the covariant gauges for the photon field results from the gauge invariance of the Lagrangian (1) A gauge transformation of the photon field,... 

Note that L2 does not explicitly depend on proper time t, since according to Eq. (3.92) r is uniquely determined by the space-time vector x and the 4-velocity u. For a better comparison between the three-dimensional formulation (Li) and the explicitly covariant formulation (L2) we have employed the velocity v = r (instead of r itself) in Eq. (3.139). Both Lagrangians Li and L2 do not represent physical observables and are therefore not uniquely determined. According to the Hamiltonian principle of least action given by Eq. (2.48), 5S = 0, they only have to yield the same equation of motion. This is in particular guaranteed if even the actions themselves are identical, i.e.. 

Now we repeat our considerations for the explicitly covariant Lagrangian L2. Due to the homogeneity of space-time, i.e., translational invariance in space and time L2 must not depend on x, and due to rotational invariance it must not depend on the 4-velocity u directly, but only on its four-dimensional length i.e., L2 = L2(m) = Analogously to the nonrelativistic... 

In analogy to our argumentation for the free particle above, we require the three-dimensional formulation of the Lagrangian L = Li r,v,t) to yield exactly the same action S as the explicitly covariant Lagrangian L2 = L2 x, u), i.e., the interaction term has to be rewritten as... 

Application of a variation of the four-dimensional path Xi(t) of the particle to the Lagrangian L2 immediately yields the covariant Euler-Lagrange equations for the charged particle. 

A Lagrangian-picture theory that is reducible to the basic Eulerian variables will be covariant with respect to the transformation (Equation 4.34). The Lagrangian... 

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