Lagrangian density interactions

In the most general case the Lagrangian density of a field suffers a reduction of symmetry at some critical value of an interaction parameter. Suppose that... 

The generalized Lagrangian density of the non-linear er-w-model in the RMF approximation used for modeling the phase of uniform nuclear matter containing interacting neutrons, protons, muons and electrons can be written as... 

As in the case of the electromagnetic self-mass, the implied dynamical mass increment is infinite unless perturbation-theory sums are truncated by a renormalization cutoff procedure. In analogy to electrodynamics, each fermion field acquires an incremental dynamical mass through interaction with the gauge field. This implies in electroweak theory that neutrinos must acquire such a dynamical mass from their interaction with the SUIT) gauge field. For a renormalized Dirac fermion in an externally determined SUIT) gauge field, the Lagrangian density is... 

The Lagrangian density for a massless fermion field interacting with the S(J(2) gauge field is... 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

The quantum mechanical Lagrangian density for a system of many particles interacting via a many-particle potential energy operator V is... 

This interaction is SU(2)l x U(1)y symmetric. Upon spontaneous symmetry breaking, the electrons acquire mass. If we use the vacuum value of the Higgs field, the Lagrangian density reduces to... 

The terms of the polynomial, approximating the Lagrangian density above the second order describe the behaviour of the interacting fields... 

This interaction Lagrangian density may depend explicitly on the space-time coordinates x and the 4-velocity u via the charge-current density T. However, as far as only the equation of motion for the electrodynamic field is concerned they do not represent dynamical variables. Lorentz invariance of this interaction term is obvious, and gauge invariance of the corresponding action is a direct consequence of the continuity equation for the charge-current density f, cf. Eq. (3.162),... 

The Lagrangian density em for the electromagnetic field is therefore given as the sum of the kinetic term 3 and the interaction term... 

We will now examine one of the most central features of the interaction between the electromagnetic field and charged matter as described by the interaction Lagrangian density int given by Eq. (3.189). As has been explicitly shown above, this form for jnt is the simplest choice compatible with all... 

FM at some density 1. One of the essential points we learned here is that we need no spin-dependent interaction at the original Lagrangian to see SSP. We can see a similar phenomenon in dealing with nuclear matter within the relativistic mean-field theory, where the Fock interaction can be extracted by way of the Fierz transformation from the original Lagrangian [11],... 

All solutions of this Hamiltonian are thereby electronic, whether they are of positive or negative energy and contrary to what is often stated in the literature. Positronic solutions are obtained by charge conjugation. From the expectation value of the Dirac Hamiltonian (23) and from consideration of the interaction Lagrangian (16) relativistic charge and current density are readily identified as... 

My investigation proceeded along the following lines [12]. The conventional coupling of an external electromagnetic potential field A to an electrical four-current density of matter is in terms of the scalar interaction Lagrangian... 

The various bilinear fermionic terms in the interaction Lagrangian are four-currents that interact with the gauge bosons. The Lagrange density... 

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