Lorentz transformation operator

Wightman, A. S., and Sohweber, S. S., Phye. Bee., 98,812 (1954). A discussion of the transformation properties of the operators and under Lorentz transformation is also included in this reference. 

For a discussion of the transformation of the field operators under improper Lorentz transformations and discrete symmetry operations such as charge conjugation, see ... 

The formalism can be carried farther to discuss the particle observables and also the transformation properties of the s and of the scalar product under Lorentz transformations. Since in our subsequent discussion we shall be primarily interested in the covariant amplitudes describing the photon, we shall not here carry out these considerations. We only mention that a position operator q having the properties that ... 

We have noted that the unitary operators U(a,A) define a representation of the inhomogeneous group. If we denote by P and AT, the (hermitian) generators for infinitesimal translations and Lorentz transformations respectively, then... 

Similar considerations lead to the transformation properties of the one-photon states and of the photon in -operators which create photons of definite momentum and helicity. We shall, however, omit them here. Suffice it to remark that the above transformation properties imply that the interaction hamiltonian density Jf mAz) = transforms like a scalar under restricted inhomogeneous Lorentz transformation... 

The vanishing of this matrix element is, in fact, independent of the assumption of current conservation, and can be proved using the transformation properties of the current operator and one-partic e states under space and time inversion, together with the hermiticity of jn(0). By actually generating the states q,<>, from the states in which the particle is at rest, by a Lorentz transformation along the 3 axis, and the use of the transformation properties of the current operator, essentially the entire kinematical structure of the matrix element of on q, can be obtained.15 We shall, however, not do so here. Bather, we note that the right-hand side of Eq. (11-529) implies that... 

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... 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

Lorentz transformations is also invariant under the combined operations of charge conjugation, C, space inversion, P, and time reversal, T, taken in any order. 

One of the central problems in the theory of P decay is the determination of the Hamilton operator of the weak interaction (O Eqs. (2.71) and O (2.72)). H should be invariant against proper Lorentz transformation, otherwise it would be possible to determine an absolute time, which is impossible according to the theory of relativity. Then, from Dirac s relativistic wave mechanics for spin 1/2 particles, it follows that there may be five classes of weak interaction terms, each transforming in a particular way under rotation and space inversion scalar (S), vector (V), antisymmetric tensor of second rank (T), axial vector (A), and pseudoscalar (P). As one cannot exclude any of these from the beginning, a linear combination of all five interactions must be considered ... 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

Quantities without any indices such as the mass m or the space-time interval ds, which are not only covariant but invariant under Lorentz transformations, are called Lorentz scalars or zero-rank tensors. They have exactly the same value in all inertial frames of reference. A very important scalar operator for both relativistic mechanics and electrodynamics is the d Alembert operator... 

Since the d Alembert operator is a Lorentz scalar, cf. Eq. (3.51), and the charge-current density fi has been shown to be a Lorentz 4-vector, it is immediately obvious that the gauge field also represents a Lorentz 4-vector and transforms according to Eq. (3.36) xmder Lorentz transformations. 

It is the fundamental Eq. (4.16), which has to obey the principle of Einstein s theory of special relativity, namely of being invariant in form in different inertial frames of reference. Hence, the choices for the Hamiltonian operator H are further limited by the requirement of form invariance of the whole equation under Lorentz transformations which will be discussed in detail in chapter 5. 

The (4x4)-matrix operator/a acts on Dirac 4-spinors and is to be determined. Owing to the operator /a the Lorentz transformation, which relates the coordinates of both inertial frames of reference IS and IS, mixes the components Yj of Y. Eq. (5.56) can be written for each component of the new state vector Y as... 

In dealing with fields that vary over time and space, we will need various differential operators. In the nonrelativistic theory of electrodynamics the gradient operator, V, and the time derivative, d/dr, are used. From our experience in the previous chapter with mixing of space and time coordinates under Lorentz transformations, we might expect these to combine in a four-space differential operator also. Indeed, in our notation. 

Comparing with (4.27), we see that we may identify ca with the velocity operator u. With this identification, we see that the velocity four-vector a can be identified with the classical velocity four-vector, which is the time derivative of the position four-vector w. We can also identify p with the inverse of the y factor arising in the Lorentz transformations. 

What is of further concern is whether the probability density is time-independent, which we expect for a bound state, and whether it is conserved under a Lorentz transformation, since this has implications for the normalization of the wave function. If ca is the velocity operator, we may write the current density for the Dirac wave function as... 

Here we recognize a situation that is quite similar to that encountered for the 50(3) rotations in section 6.4. We have transformations that conserve the length of the four-vector, and which may be expressed as matrices operating on the set of basis vectors. We may in fact proceed in a completely analogous manner to the SO(3) case. Thus, an infinitesimal Lorentz transformation— that is, v/c 1—may be written as... 

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. 

Requiring these order parameters to transform in a Lorentz-covariant way, we are led to a particular basis of 4 x 4 matrices , which was recently derived in detail (Capelle and Gross 1999a). The resulting order parameters represent a Lorentz scalar (one component), a four vector (four components), a pseudo scalar (one component), an axial four vector (four components), and an antisymmetric tensor of rank two (six independent components). This set of 4 x 4 matrices is different from the usual Dirac y matrices. The latter only lead to a Lorentz scalar, a four vector, etc., when combined with one creation and one annihilation operator, whereas the order parameter consists of two annihilation operators. 

The right-hand side of this equation is a mere cosine function as shown in Figure 6.6d, and its Fourier transform gives a delta function as shown in Figure 6.6e. This means that in an ideal case the assumed true bandshape is recovered by the above series of computations. What is needed in a practical case is an operation to narrow the bandwidth of the Lorentz profile to an appropriate degree. In more general terms, FSD is an operation to narrow (deconvolve) the bandwidth of a band by computations involving the (inverse) Fourier transform of the bandshape function itself (FSD). 

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