Space Inversion

The effect of space inversion I is to change the sign of all spatial coordinates 

This corresponds to multiplication of the position vector by the matrix —13 

Our experience with inversion is related to the nonrelativistic case and operations in R3. We can always apply the operator in the nonrelativistic form, but our aim is to find operations that leave the Dirac Hamiltonian invariant, and there is no guarantee that the nonrelativistic inversion operator will do this. However, the operator we are looking for should have the same effect as the familiar inversion when applied only to R3. We may therefore assume that the inversion operator we are looking for may be written as the product of one part that acts only in R3 in the manner described in (6.99) and one part that does not act in R3, but which may affect other coordinates. We write the relativistic inversion operator as 

p is only affected by the Ir part of the operator, and thus 

The Other terms are unaffected by operations on the spatial coordinates, but may be affected by the operator a. Therefore 

---

The discussion at the beginning of this section, when coupled with the fact that the observers 0 and O agree to describe bodily the same state by the same state vector, has exhibited the invariance of quantum electrodynamics under space inversion in the Heisenberg-type description. 

Let us next adopt the Schrodinger-type description. The statement that quantum electrodynamics is invariant under space inversion can now be translated into the statement that there exists a unitary operator U(it) such that... 

Invariance of Quantum Electrodynamics under Discrete Transformations.—In the present section we consider the invariance of quantum electrodynamics under discrete symmetry operations, such as space-inversion, time-inversion, and charge conjugation. 

As indicated at the beginning of the last section, to say that quantum electrodynamics is invariant under space inversion (x = ijX) means that we can find new field operators tfi (x ),A v x ) expressible in terms of fj(x) and A nix) which satisfy the same equations of motion and commutation rules with respect to the primed coordinate system (a = igx) as did tf/(x) and Av(x) in terms of x. Since the commutation rules are to be the same for both sets of operators and the set of realizable states must be invariant, there must exist a unitary (or anti-unitary) transformation connecting these two sets of operators if the theory is invariant. For the case of space inversions, such a unitary operator is... 

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... 

Particle-Antipartide Conjugation.—If quantum electrodynamics is invariant under space inversion, then it does not matter whether we employ a right- or left-handed coordinate system in the description of ptnely electrodynamio phenomena. To speak of right and left is, an arbitrary convention in a worlcl ip which only electrodynamics operates. 

As in the ease of space inversion, one readily verifies that... 

From the invariance of the theory under space inversion, it follows that the axial vector and tensor amplitudes transform as follows ... 

Space inversion in quantum electrodynamics, 679 Spatial operators... 

It was believed for a long time that the fundamental laws of nature are invariant under space inversion, and hence the conservation of space inversion symmetry (P) is a universally accepted principle. The nonconservation of this symmetry was discovered experimentally by Wu and co-workers in the (3 decay of 60Co in... 

On the other hand, the permanent EDM of an elementary particle vanishes when the discrete symmetries of space inversion (P) and time reversal (T) are both violated. This naturally makes the EDM small in fundamental particles of ordinary matter. For instance, in the standard model (SM) of elementary particle physics, the expected value of the electron EDM de is less than 10 38 e.cm [7] (which is effectively zero), where e is the charge of the electron. Some popular extensions of the SM, on the other hand, predict the value of the electron EDM in the range 10 26-10-28 e.cm. (see Ref. 8 for further details). The search for a nonzero electron EDM is therefore a search for physics beyond the SM and particularly it is a search for T violation. This is, at present, an important and active held of research because the prospects of discovering new physics seems possible. 

After the discovery of the combined charge and space symmetry violation, or CP violation, in the decay of neutral mesons [2], the search for the EDMs of elementary particles has become one of the fundamental problems in physics. A permanent EDM is induced by the super-weak interactions that violate both space inversion symmetry and time reversal invariance [11], Considerable experimental efforts have been invested in probing for atomic EDMs (da) induced by EDMs of the proton, neutron, and electron, and by the P,T-odd interactions between them. The best available limit for the electron EDM, de, was obtained from atomic T1 experiments [12], which established an upper limit of de < 1.6 x 10 27e-cm. The benchmark upper limit on a nuclear EDM is obtained from the atomic EDM experiment on Iyt,Hg [13] as d ig < 2.1 x 10 2 e-cm, from which the best restriction on the proton EDM, dp < 5.4 x 10 24e-cm, was also obtained by Dmitriev and Senkov [14]. The previous upper limit on the proton EDM was estimated from the molecular T1F experiments by Hinds and co-workers [15]. 

The space inversion transformation is x —> —x and the corresponding operator on state vector space is called the parity operator (P). The parity operator reverses... 

As mentioned earlier, heavy polar diatomic molecules, such as BaF, YbF, T1F, and PbO, are the prime experimental probes for the search of the violation of space inversion symmetry (P) and time reversal invariance (T). The experimental detection of these effects has important consequences [37, 38] for the theory of fundamental interactions or for physics beyond the standard model [39, 40]. For instance, a series of experiments on T1F [41] have already been reported, which provide the tightest limit available on the tensor coupling constant Cj, proton electric dipole moment (EDM) dp, and so on. Experiments on the YbF and BaF molecules are also of fundamental significance for the study of symmetry violation in nature, as these experiments have the potential to detect effects due to the electron EDM de. Accurate theoretical calculations are also absolutely necessary to interpret these ongoing (and perhaps forthcoming) experimental outcomes. For example, knowledge of the effective electric field E (characterized by Wd) on the unpaired electron is required to link the experimentally determined P,T-odd frequency shift with the electron s EDM de in the ground (X2X /2) state of YbF and BaF. 

Solvent properties, transition state trajectory, future research issues, 232-233 Space inversion symmetry (P) ab initio calculations, 253—259 barium fluroide molecules, 256-259 ytterbium molecule, 254—256 electric dipole moment search, 241-242 nonconservation, 239—241 Spatial neighbor tables, Monte Carlo heat flow simulation, 68—70... 

True chirality is exhibited/possessed by systems that exist in two distinct enantiomeric states that are interconverted by space inversion, but not by time reversal combined with any proper spatial rotation. 

Note that the original definition has now evolved into a dichotomous classification Truly chiral systems exist in two distinct enantiomorphous states that are interconverted by space inversion but not by time reversal combined with any proper spatial rotation, whereas falsely chiral systems exist in two distinct enantiomorphous states that are interconverted by space inversion or by time reversal combined with any proper spatial rotation.34- 35 The process of time reversal, represented by the operator T, is the same operation as letting a movie film run backward. The act of inversion [i.e., time reversal] is not a physical act, but the study of the opposite chronological order of the same items. 38... 

Figure 4. A translating spinning cylinder. The polar vector in the rotation-translation (screw displacement) corresponds to the direction of translation and the axial vector to the direction of spin. Time reversal (7) does not change the sense of chirality of homomorphous systems (a) and (b) in terms of the helicity generated by the product of the two vectors, (a) and (b) are both right-handed. Space inversion (P) of (a) yields a left-handed system (c), the enantiomorph of (a). Time reversal of (a), followed by rotation of (b) by 180° (Rn) about an axis perpendicular to the cylindrical axis, yields (d), a homomorph of (a). Space inversion of (d) brings us back to (c).

The next section (Sect. 2) is devoted to a lengthy discussion of the molecular hypothesis from the point of view of quantum field theory, and this provides the basis for the subsequent discussion of optical activity. Having used linear response theory to establish the equations for optical activity (Sect. 3), we pause to discuss the properties of the wavefunctions of optically active isomers in relation to the space inversion operator (Sect. 4), before indicating how the general optical activity equations can be related to the usual Rosenfeld equation for the optical rotation in a chiral molecule. Finally (Sect. 5), there are critical remarks about what can currently be said in the microscopic quantum-mechanical theory of optical activity based on some approximate models of the field theory. 

In the particular case of optical activity I can describe the experiment by choosing % to be the electric polarization operator P(x, t) for the material medium, while tot refers to the combined system of matter interacting with the polarized light beam. Note that [ tot> P] = 0, where P is the space-inversion operator (see40 and Sect. 4). Then the mean value,... 

In order to make the connection with the usual discussion of optical rotation based on the Rosenfeld equation for a molecule49, one must express the exact states ip, of the chiral medium in terms of the states of the elementary excitations in the system, using the machinery of quantum field theory discussed in Sect. 2. Before considering this problem however it is instructive to consider first the role of the space-inversion operator P in optical activity. 

Let P be the operator describing space-inversion the identity, f, and P together form an Abelian group, <8(c), isomorphic to 212, with the composition law ... 

---