Transform properties

Transformation properties and Hamiltonian for tetraatomic systems J. Phys. Chem. A 101 6368-83... 

The vanishing of the YM field intensity tensor can be shown to follow from the gauge transformation properties of the potential and the field. It is well known (e.g., Section II in [67]) that under a unitary transfoiination described by the matrix... 

The spin operator S is an irredueible tensor of rank one with the following transformational properties... 

These definitions arise from the transformation properties of vectors and can be summarized as follows If in the transfonuation of the coordinate system to another system quantities Aj, A2, ., A trans-... 

These same rotation matriees arise when the transformation properties of spherieal harmonies are examined for transformations that rotate eoordinate systems. For example,... 

It is possible to assume other transformation properties for k. For example, for some purposes it may be more desirable to attribute strainlike properties obeying a transformation law like (A. 19), in which case the equations of this section will take a somewhat different form. Of course, k may be taken to be comprised of a number of such tensors, and it is not difficult to extend the theory to include a number of indifferent scalars and vectors, if desired. 

Again, other transformation properties might be assumed, and k may be taken to be comprised of a number of such tensors, or to include a number of invariant scalars or vectors. 

Because the ID unit cells for the symmorphic groups are relatively small in area, the number of phonon branches or the number of electronic energy bands associated with the ID dispersion relations is relatively small. Of course, for the chiral tubules the ID unit cells are very large, so that the number of phonon branches and electronic energy bands is also large. Using the transformation properties of the atoms within the unit cell transformation... 

Hence we may conclude for a vibration to be active in the infrared spectrum it must have the same symmetry properties (i.e. transform in the same way) as, at least, one of x, y, or z. The transformation properties of these simple displacement vectors are easily determined and are usually given in character tables. Therefore, knowing the form of a normal vibration we may determine its symmetry by consulting the character table and then its infrared activity. 

In quantum mechanics, angular momenta other than orbital make their appearance. Their structure is not revealed by the simple considerations leading to (7-8). That formula, in fact, arises also from the general transformation properties of vectors under rotation, as will now be shown. 

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

We next inquire as to the transformation property of the quantity ip(z) under a homogeneous Lorentz transformation... 

Transformation properties of Dirac spinors in particular under inversions Marshak, R. E., and Sudarshan, E. C. G., Introduction to Elementary Particle Physics, Interscience Publishers, Inc., New York, 1961. 

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. 

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 ... 

In order to establish the transformation properties of the one-particle states, let us obtain the transformation properties of the in operators. 

This is verified by applying U(a,A) to the left and U(a,A) l to the right of both sides of Eqs. (11-142) and (11-97), mid making use of the transformation properties of (11-184) and (11-185) of the fields. Equations (11-241) and (11-242) were to be expected. They guarantee that four-vector, respectively. We are now in a position to discuss the transformation properties of the one-particle states. Consider the one-negaton state, p,s,—e>. Upon taking the adjoint of Eq. (11-241) and multiplying by ya we obtain... 

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... 

These transformation properties can be transcribed into transformation properties of state vectors in a Schrodinger-type description. The in-fields will satisfy equations similar to the above, e.g. 

Note the minus sign on the right side of Eq. (11-276), stemming from the relation (11-274). These transformation properties imply the following transformation rule of a one-negaton state under space inversion... 

Note that the above transformation properties for the operators force the current operator to transform as expeoted under is, namely like a vector. Since... 

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

The above transformation properties of the current operator make quantum electrodynamics invariant under the operation Ue, usually called charge conjugation, provided... 

More generally, let us define the transformation properties of the Heisenberg field to be... 

The choice of ijr 2 — 1, together with the antiunitary character of U(it), guarantees the invariance of the equal time commutation rules under U(it). With these definitions of the transformation properties of the spin field operators one verifies that... 

Grouping the r(0 into sets with well-defined transformation properties, we can write Eq. (11-402) more explicitly in the form... 

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... 

Finally, this type of analysis can be carried out for any point in the Brillouin zone such that by using the transformation properties of spin waves and the character tables, one may obtain the spin-wave band structure throughout the zone. 

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