Perturbation theory Lagrangian

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

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... 

These parameters can be evaluated using effective Lagrangian and Chiral Perturbation Theory [18,19,20,21,22] which is mathematically equivalent to QCD [19,21]. At present time, the value ao — a2 has been determined within 2% [23], The QCD Lagrangian and effective Lagrangians are determined by Lorentz invariance, P and C-invariance and by the chiral symmetry. For this reason, the measurement of do — < 21 provides an opportunity to check our understanding of the chiral symmetry breaking of QCD. 

The expressions derived for the molecular properties in the previous section are of a rather general and perhaps somewhat abstract character. For a given variational wave function, the explicit expressions for the molecular properties are obtained by substituting in Eqs. 18 to 21 the detailed form of the energy functional (x A) for a nonvariational wave function, we first express the energy as a variational Lagrangian and then proceed in the same manner. We shall not discuss the detailed expressions for the derivatives here, referring instead to special reviews [1]. Still, to illustrate the physical contents of Eq. 18 and Eq. 21, we shall now see how these expressions are related to those of standard time-independent perturbation theory. 

Frequency-dependent higher-order properties can now be obtained as derivatives of the real part of the time-average of the quasi-energy W j- with respect to the field strengths of the external perturbations. To derive computational efficient expressions for the derivatives of the coupled cluster quasi-energy, which obey the 2n-(-1- and 2n-(-2-rules of variational perturbation theory [44, 45, 93], the (quasi-) energy is combined with the cluster equations to a Lagrangian ... 

This is an important quantity. Our first development of it followed the above, derivative argument [ 103,104,109]. This simply means we use first-order perturbation theory with the CC reference as the unperturbed problem to get the first-order energy correction—the gradient. It was also suggested from a different field as a generalization of CC theory [110, 111], but in chemistry we consider energies and their derivatives (properties) to be synonymous as above [111]. Finally, it can also be deduced as a consequence of a Lagrangian multiplier constraint [112]. 

In a field-theoretic picture, the interaction between mesons and baryons can be described by effective Lagrangians, The NN interaction can then be derived in terms of field-theoretic perturbation theory. The lowest order (that is, the second-order in terms of meson-baryon interactions) are the one-boson-exchange diagrams (Fig. IX which are easy to calculate. 

If we take the form of C given in (3.2.9) as our quantum Lagrangian and we attempt to do perturbation theory, all propagators will look sensible, with poles corresponding only to the physical particles. For this reason the chosen gauge is called a unitary gauge U gauge). 

The relation (4.2.17) follows directly from the Lagrangian, and is therefore true in lowest order perturbation theory. It could get altered in higher order perturbation theory, especially as a consequence of renormalization effects. As regards the Higgs scalar H which survives and becomes massive, it will as usual have a mass (3.2.10)... 

Similar transition expectation values can also be defined for other non-variational methods like Mqller-Plesset perturbation theory, where one defines a Lagrangian by adding the equations for the correlation coefficients as extra conditions multiplied with Lagrangian multipliers to the respective MP energy expression (Hattig and Hefi, 1995 Aiga and Itoh, 1996). 

The renormalisation constants Z, Z2, Z3 and dm have to be understood as functions of the finite physical charge e and mass m of the electrons which can be constructed order by order in the perturbation series. It is important to notice that these constants are uniquely determined by vacuum QED without any external potential. They do not depend on the specific external potential present. If one bases the perturbation expansion on the Lagrangian (A.49) all Greens and n-point functions of the theory (defined in terms of the physical fields tj/ and T ) are finite. 

Presumably the change of variables (3.1.8) cannot alter the physics if the problem is solved exactly. However, if perturbation methods are used in the quantum theory, this need not be true. For example, while in (3.1.9) it might be sensible to take the kinetic energy and mass terms as the unperturbed Lagrangian q, it would be disastrous to do so in (3.1.1) because of the negative mass squared terms. 

If higher order perturbative effects are calculated using the Lagrangian couplings given in (5.1.1) and (5.1.3) in which the parameters e and sin iv are considered as fixed numbers, many diagrams yield infinite answers and the theory has to be renormalized. 

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