Operator gauge transformed

Hence, the method of Mead and Truhlar [6] yields a single-valued nuclear wave function by adding a vector potential A to the kinetic energy operator. Different values of odd (or even) I yield physically equivalent results, since they yield (< )) that are identical to within an integer number of factors of exp(/< )). By analogy with electromagnetic vector potentials, one can say that different odd (or even) I are related by a gauge transformation [6, 7]. 

The derivation of Eq. (218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d Alembert wave equation is replaced by an 0(3) covariant derivative. The U(l) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically. The vacuum currents present in Eq. (218) can be computed from the right-hand side of the wave equation (212), and these vacuum currents follow from local gauge invariance. 

If the physical results are to remain unchanged under a unitary transformation, it is necessary to transform the operators as well as the state functions (eqns (8.28) and (8.29)). A gauge transformation has no effect on a coordinate operator but the momentum operator p is changed into p + [e/c) x- Thus, to maintain gauge invariance in properties determined by the momentum operator, p must be replaced by a new operator. In the presence of an electromagnetic field or just a magnetic field, the momentum operator is replaced by the expression... 

In principle, a chemical shift calculation represents a perturbation theory, because of the presence of an external field Bz and magnetic moments due to the dipole character of nuclei. Therefore, perturbations to the Hamiltonian and the wave function have to be considered. The next important point is that the origin of the vector potential Az is not fixed due to the relation Bz = rot Az- Any change of the gauge origin Rq should not change any measurable observable. Therefore, a gauge transformation of the wave function 1%) and Hamilton operator h is essential... 

D.H. Kobe, K.-H. Yang, Gauge transformation of the time-evolution operator, Phys. Rev. A 32 (1985) 952. 

Before we can start with the discussion of time-dependent perturbation theory in the form of response theory, we need to introduce an alternative formulation of quantum mechanics, called the interaction or Dirac representation. In general, several representations of the wavefunctions or state vectors and of the operators of quantum mechanics are equivalent, i.e. valid, as long as the expectation values of operators ( 0 I d I o) or inner products of the wavefunctions ( o n) are always the same. Measurable quantities and thus the physics are contained in the expectation values or inner products, whereas operators and wavefunctions are mathematical constructs used in a particular formulation of the theory. One example of this was already discussed in Section 2.9 on gauge transformations of the vector and scalar potentials. In the present section we want to look at a transformation that is related to the time dependence of the wavefunctions and operators. 

For approximate theories, however, different results are always obtained with different perturbation operators related by gauge transformations. The use of a finite basis set implies that operators are represented as finite matrices and, consequently, the commutator relations no longer hold. As the basis set quality is increased, the commutators converge to the exact values. Gauge invariance thus is recovered in the limit of a complete basis set, provided that the equation of motion O Eq. 5.46, is fulfilled. While this is the case in theories with fuHy variational orbitals such as DFT, HF, and MCSCF, it is not the case in approximate theories with fixed (nonvariational) orbitals such as Cl and CC. 

This translation of the operator can also be achieved by a particular type of gauge transformation of the magnetic vector potential, explaining why the term gauge-origin is often used for O in the context of magnetic properties. 

What is shown in Fig. 1 is the oscillator strength ig the "length gauge. As has been described in detail by Grant, the relativistic transition operators which correspond in the nonrelativistic limit to the length and velocity operators are related to each other by a gauge transformation of the photon operator. The MCHF proceedure is not gauge invariant, basically because a truncation has been carried out in the sum over creation and annihilation operators in the definition of the many-electron... 

The matter field operators, although denoted by the same symbol, should not be confused with those defined in the Coulomb gauge by the equations of motion (11-47). The same is true for the in, out operators. The two are related by a transformation of the form... 

Quantization of radiation field in terms of field intensity operators, 562 Quantum electrodynamics, 642 asymptotic condition, 698 gauge invariance in relation to operators inducing inhomogeneous Lorentz transformations, 678 invariance properties, 664 invariance under discrete transformations, 679... 

The transformation of the relativistic expression for the operator of magnetic multipole radiation (4.8) may be done similarly to the case of electric transitions. As has already been mentioned, in this case the corresponding potential of electromagnetic field does not depend on the gauge condition, therefore, there is only the following expression for the non-relativistic operator of Mk-transitions (in a.u.) ... 

Non-Abelian electrodynamics has been presented in considerable detail in a nonrelativistic setting. However, all gauge fields exist in spacetime and thus exhibits Poincare transformation. In flat spacetime these transformations are global symmetries that act to transform the electric and magnetic components of a gauge field into each other. The same is the case for non-Abelian electrodynamics. Further, the electromagnetic vector potential is written according to absorption and emission operators that act on element of a Fock space of states. It is then reasonable to require that the theory be treated in a manifestly Lorentz covariant manner. 

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