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CP violation and molecular physics
22 April 1994
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 22 1 ( 1994) 3 1l-3 16
CP violation and molecular physics L.D. Barron Chemistry Department, The University, Glasgow G12 8QQ, UK Received 29 April 1993; in final form 6 January 1994
Abstract The proof that a particle and its antiparticle have identical rest mass, which is based on the CPT invariance of the Hamiltonian, can be adapted to show that the CP enantiomers of a chiral molecule remain degenerate even in the presence of CP violation. This reinforces an earlier suggestion that the force responsible for CP violation exhibits false chirality with respect to CP enantiomorphism and leads naturally to the idea that the CP violation observed in the neutral K-meson system is analogous to a special type of chemical catalysis since it affects the rates of certain particle-antiparticle interconversion pathways (via a breakdown in microscopic reversibility) without affecting the initial and final particle energies and hence the equilibrium thermodynamics. CP violation of this type is unlikely to have any direct manifestations in molecular physics.
1. Introduction
The fundamental discrete symmetries of parity (P) , time reversal (T) and charge conjugation (C), and their violations in certain situations, are central to modern elementary particle physics (see e.g. refs. [ l31). In the realm of atomic and molecular physics, interest in symmetry violation has centred on parity violation induced by the weak neutral current interaction, manifestations of which are now observed routinely in polarized atomic spectroscopy experiments (see e.g. refs. [ 4,5 ] ). Since the combined CP symmetry is known to be violated in certain elementary particle processes [ 6 1, the possibility that CP violation might appear in atomic and molecular physics has also attracted some discussion [ 7,8] which has of necessity been speculative on account of the current incomplete understanding of this phenomenon [9-l 11. This Letter presents a new theorem conceming CP violation in chiral molecules, together with an analogy between CP violation in the neutral K-meson system and chemical catalysis, which emphasise
that CP violation falls within the conceptual framework of false chirality [ 12,13 1. These new insights provide a cornerstone for future discussions of CP violation in molecular physics.
2. True enantiomers Much has been made of parity violation in the special case of chiral molecules where the degeneracy of the mirror-image enantiomers is lifted (see e.g. refs. [ 4,14-l 8 ] ). Some years ago, I showed that the true enantiomer of a chiral molecule (i.e. the ‘opposite’ object with identical energy to the original) is the molecule with the opposite absolute configuration but composed of antiparticles, being generated from the original by the combined CP operation [ 19-22 ] ; this means that a chiral molecule is associated with two distinct pairs of true enantiomers (Fig. 1) . However, contrary to the impression given recently by Quack [ 81, I never intended that the term ‘enantiomers’ not be used for an ordinary optically active molecule and
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDi0009-2614(94)00253-M
312
L.D. Barron /Chemical Physics Letters 221(1994) 311-316 P
M
c,
h-l=
where O= CPT is the associated operator. Taking the matrix element of both sides between any two particle states labelled a and b, we have (b]H]a)=(blO-‘HSZ]a) =(~bl~~-‘Hnla)*=(~lH~la)*.
i
c--,
P
-m
M
Fig. 1. The two pairs of true (strictly degenerate) enantiomers (M, IP and M*, M) of a chiral molecule that are interconverted by CP.
its mirror image in ordinary stereochemistry; I was making the academic point that these are not strict opposites (I use the terminology ‘P-enantiomers’ and ‘CP-enantiomers’ when discussing these fundamental aspects [ 18 1, but not in ordinary stereochemistry). Quack then goes on to say that, since CP itself is known to be violated, we should allow in principle for the degeneracy of even CP enantiomers to be lifted by some CP violating interaction in molecules [ 8 1. My earlier papers also allowed for this possibility; but I now realize that, provided CPT invariance holds, the idea is wrong since, as proved below, CP enantiomers remain strictly degenerate even in the presence of CP violation.
3. CP enantiomers are always true The CPT theorem of relativistic quantum field theory states that, even if one or more of the operations C, P and T is violated, the combined operation of CPT is always conserved [ 23 1. This theorem can be proved in great generality with only mild assumptions [ 241. There are three important consequences [ 1,2,11]: the rest mass of a particle and its antiparticle are equal; the particle and antiparticle lifetimes are the same (even though decay rates for individual channels may not be equal); and the electromagnetic properties such as charge, magnetic moment, etc. are opposite. We are particularly interested in the mass, and hence energy, equality which is proved in the followingway [1,2,11]. Invariance of any Hamiltonian H under CPT is expressed by Q-‘HQ=H,
(1)
(2)
The second equality follows from the antiunitary nature of 8 associated with the time reversal operator contained within it (so that (8y/] @r) = (v/I 9) *). When applied to the state of a particle at rest, the operator g has the following effect:
Qla>=tla>,
(3)
where a labels the corresponding antiparticle state and
(4)
Taking a and b to denote the state of a single particle at rest, (4) becomes the real expectation value (a]H]a)
= (i%lHla)*= (alHI%)
.
(5)
Hence if i contains all the internal interactions which contribute to the particle mass, the mass equality of particle and antiparticle is proved. If the particle has spin, the spin angular momentum vector of the antiparticle will point in the opposite direction (because of T in a); but this feature is immaterial for a free particle because the condition of Lorentz invariance required for the derivation of the CPT theorem covers rotational symmetry so the original spin angular momentum direction can always be restored without affecting the energy. We now come to an important point. The standard derivation given above assumes that the P part of 52 does not alter the particle itself [ I] (although if the particle is not at rest its helicity will be reversed by P). But if a is now taken to denote the state of a stationary chiral molecule, we must allow for P to generate the mirror-image enantiomeric state denoted a”. Hence (3) must be generalized to
Qla>=TlaX>,
(6)
so that ( 5) becomes (alHla)=(aX]Hla”),
(7)
which proves the rest-mass, and hence energy, equiv-
L.D. Barron /Chemical PhysicsLetters221(1994) 31 l-316
alence of the CP enantiomers of a chiral molecule. Since only thefull CPT invariance of the Hamiltonian is invoked in the derivation, this result applies if any of C, P and T either separately or in combination are violated. Hence the CP enantiomers of a chiral mol-
ecule remain strictly degenerate even in the presence of CP violation. If the molecule has non-zero spin angular momentum associated with an odd number of electrons (or rotational angular momentum if in the gas phase) the result is not affected because the argument in the previous paragraph still applies. A similar extension of the proof of the lifetime equality of particle and antiparticle from CPT invariance [ 1,2,11] can be easily developed, if required, to take account of the fact that a chiral molecule is not in a stationary state of its Hamiltonian [ 17,18,21]. The above analysis shows that an energy difference between the CP enantiomers of a chiral molecule requires CPT violation. Hence any attempt to measure this energy difference would in fact constitute a search for both CP and CPT violation, not just CP violation as previously supposed [ 8 1. Although not feasible at present due to the current inaccessibility of antimolecules, such measurements are nonetheless interesting to contemplate in view of the recent proposal to use ultra high resolution spectroscopy to compare the lS-2s energy intervals in atomic hydrogen and antihydrogen to one part in 1O’* (which should soon be possible) as a test of CPT invariance to very high precision [ 25 1.
4. False chirality and CP violation The concept of false chirality was originally introduced to distinguish time-invariant P-enantiomorphism (‘true chirality’) from time-noninvariant P-enantiomorphism (‘false chirality’) [ 12,13 1. An example of the former is a conventional pair of mirror-image chiral molecules where two distinguishable enantiomers are interconverted by P but not by T. An example of the latter is a system of collinear electric and magnetic fields where two distinguishable enantiomers (the parallel and antiparallel arrangements) are interconverted not only by P but also by T so that it is PT invariant overall. Although not referred to as such, a version of false chirality also
313
arises in the anyon theory of high temperature superconductivity [ 26 ] where a spinning system in two dimensions breaks both P and T separately but again is invariant under the combined PT operation (parity has a different effect in two dimensions than in three since it becomes simply a reflection in only one axis t271). It has been pointed out previously that any force responsible for CP violation exhibits an analogue of false chirality with respect to CP enantiomorphism [ 18,28 1. This is because CP-violating forces always come in pairs that are interconverted by CP, with CPT invariance guaranteeing that the two distinct CP-enantiomeric forces are also interconverted by T. Violation of CP arises because only one of a CP-enantiomeric pair of forces is found in our universe. Having shown in section 3 that, if CPT invariance holds, no CP-violating force can lift the degeneracy of CP enantiomers (be they particle-antiparticle pairs or a chiral molecule and its mirror image made of antiparticles), the analogy is now satisfyingly complete since a falsely chiral P-enantiomorphous influence such as collinear electric and magnetic fields also cannot lift the degeneracy of conventional P-enantiomers such as a chiral molecule and its mirror image.
5. CP violation and chemical catalysis Violation of CP symmetry was first observed in 1964 in certain decay modes of the neutral K-meson [ 6 1. Although unequivocal, the effects are very small and to date have not been observed in any other system. On account of the CPT theorem, these experiments are also widely regarded as indirect observations of T violation (which has not yet been observed directly). One manifestation of CP violation is the following decay rate asymmetry of the long-lived neutral K-meson, the I&_[ 1,2,9- 111: d_ rate (KL+IC-e,+V~) x l 00648 rate (KL+x+e;O,) * .
As indicated, KL can decay into either positive pions A+ plus left helical electrons ep plus right helical antineutrinos &,;or into negative antipions A- plus right helical positrons e: plus left helical neutrinos vp. Since these two sets of decay products are intercon-
314
L.D. Barron I Chemical Physics Letters 221(1994) 31 l-316
verted by CP, the observed decay rate asymmetry is a manifestation of CP violation. If we naively incorporate these two decay processes into the following single ‘chemical equilibrium’ scheme involving the intermediate KL,
(9) we would say that (8) corresponds to an asymmetry in the rate constants for decay of KL into the two sets of CP-enantiomeric products, i.e. /q#@. This establishes a parallel with the absolute asymmetric synthesis which I have shown might be induced by a falsely chiral influence such as collinear electric and magnetic fields [ 18,281. As pictured in Fig. 2 for a unimolecular process in which an achiral molecule R generates a chiral molecule M or its enantiomer M +, the special feature is a breakdown in microscopic reversibility since the forward and backward potential energy profiles for reaction of a given enantiomer are different: this arises because the falsely chiral influence breaks P and T separately. However, the deeper principle of enantiomeric microscopic reversibility still holds because the influence preserves the combined PT invariance which is manifest here in the form of identical potential energy profiles for the forward and backward enantiomeric asymmetric synthesis reactions, i.e. for R-M” and M+R. Clearly a falsely chiral influence acts as a special type of catalyst in the chemical sense since it modi-
I
M'
M Reaction coordinate
Fig. 2. Potential energy profiles for enantiomeric reactions in the presence of a falsely chiral (time-noninvariant enantiomorphous) influence.
fies potential energy barriers to change relative rates of formation of enantiomeric products without affecting the relative energies of reactants and products (remember that a falsely chiral influence does not lift the degeneracy of P-enantiomeric chiral molecules) so that the position of the equilibrium is not changed [ 29 1. Since K,_and its two sets of decay products are the equivalents with respect to CP of R, M and MU with respect to P, we can conceptualize its decay rate asymmetry as arising from a breakdown in microscopic reversibility due to a time-noninvariant CP enantiomorphism in the forces of nature. The analogy is completed by the fact that in both cases the asymmetries cancel out when summed over all possible channels at true thermodynamic equilibrium [ 281. Since the CP-violating interaction responsible for the decay rate asymmetry of the KL does not lift the degeneracy of the two sets of CP-enantiomeric products, itsfunction is analogous to that of a special type of chemical catalyst.
6. CP violation in molecular physics
The long-lived and short-lived K-meson species KL and K, the decay modes of which show CP violation, are peculiar in that, being certain particle-antiparticle superposition states (which Wigner [ 301 has pointed out are analogues with respect to CP of the definite parity states 2 - ‘I2( vL ? VR) of a chiral molecule involving superpositions of the left- and righthanded P-enantiomeric states yL and @&,but slightly mixed with each other through CP violation), they constitute observable states bridging the worlds of matter and antimatter. As the equilibrium (9) suggests, these superposition states can be thought of as intermediates in particle-antiparticle transformation pathways. We might be tempted to deduce from this that CP violation and the associated breakdown in microscopic reversibility of the type observed in the neutral K-meson system could occur in principle in processes involving transformations between a chiral molecule made of matter and its currently inaccessible mirror image made of antimatter (i.e. molecular CP enantiomers), with an analogue of the K,_ or Ks involving molecule-antimolecule superposition states as an intermediate. Unfortunately, there are several fundamental impediments to this idea: for
L.D. Barron /Chemical Physics Letters 221(1994) 311-316
example, such molecule-antimolecule transformations would require a gross violation of the law of baryon number conservation which does not arise in the K-meson system because mesons have baryon number zero; and strange particles like K-mesons are probably required because specific strangeness quantum number changes are essential requirements of the current ‘superweak’ and ‘milliweak’ theories [ 9,101 of CP violation in these systems. Hence the type of CP violation observed in the Kmeson system seems unlikely to have any direct manifestations in molecular physics even if antimolecules were accessible. However, if CP and hence T violation could somehow infiltrate into the world of ordinary atoms and molecules it might, as recently suggested [ 8 1, be observed indirectly in the form of a breakdown in microscopic reversibility in molecular dynamics. And of course we can still attempt to mimic CP violation by using a falsely chiral influence (such as collinear electric and magnetic fields) to induce a breakdown in microscopic reversibility in certain processes involving ordinary chiral molecules. The necessity for particle-antiparticle processes which appears to prevent manifestations of CP violation appearing directly in atomic and molecular physics can be avoided if T is violated directly. A central example of this is a permanent electric dipole moment in a definite parity atomic or molecular state which requires simultaneous violation of both T and P and which, despite much experimental effort, has not yet been detected (see e.g. refs. [4,9-l 1,16,3 1] ). Apart from involving only ordinary matter, this situation appears to be fundamentally different from that of the neutral K-meson system since a stationary state, rather than a process, is being studied. Finally it is not widely appreciated that, as proved by Aharony [ 321, Boltzmann’s H-theorem and hence also the second law depend on unitarity rather than microscopic reversibility and so still hold even if the particles making up a system are subject to forces that violate T-invariance (although the time dependence of the H-function could be modified during the approach to equilibrium). This serves to reinforce Sachs view that there is no direct connection between the ‘arrow of time’ (the irreversibility of macroscopic phenomena) and the physics of time reversal [ 111. It also casts light on the speculation that, while it is not necessary to invoke T violation to ex-
315
plain irreversibility in statistical mechanical molecular chaos and the macroscopic world generally, an explanation based on T violation cannot be ruled out [8]: it appears that, at most, T violation can only constitute a small perturbation to the rate at which equilibrium is attained.
Acknowledgement
I thank Professor M. Quack for sending me a copy of ref. [ 81 in advance of publication and for much stimulating correspondence, and Dr. C.D. Froggatt, Dr. L. Hecht and Dr. D.G. Sutherland for discussion.
References [ I] T.D. Lee, Particle physics and introduction to field theory (Harwood, New York, 198 1) [2] W.M. Gibson and B.R. Pollard, Symmetry principles in elementary particle physics (Cambridge Univ. Press, Cambridge, 1976). [3] C.D. Froggatt and H.B. Nielsen, Origin of symmetries (World Scientific, Singapore, 199 1). [4] LB. Khriplovich, Parity nonconservation in atomic phenomena (Gordon and Breach, New York, 199 1). [ 51 M.A. Bouchiat, in: Atomic physics, Vol. 12, eds. R.R. Lewis and J.C. Zom (American Institute of Physics, 1991) p. 399. [6] J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay, Phys. Rev. Letters 13 (1964) 138. [ 7 ] AS. Gamy, ix Origins of optical activity in nature, ed. D.C. Walker (Elsevier, Amsterdam, 1979) p. 245. [8] M. Quack, J. Mol. Struct. 292 (1993) 171. [ 91 L. Wolfenstein, ed., CP violation (North-Holland, Amsterdam, 1989 ) . [ 10 ] C. Jarlskog, ed., CP violation (World Scientific, Singapore, 1989). [ 111 R.G. Sachs, The physics of time reversal (Univ. of Chicago Press, Chicago, 1987). 17 (1986) 189. 112 L.D.Barron,Chem.Soc.Rev. t13 i L.D. Barron, Chem. Phys. Letters 123 (1986) 423. 114 R.A. Hegstrom, D.W. Rein and P.G.H. Sandars, J. Chem. Phys. 73 (1980) 2329. 115 R.A. Harris, in: Quantum dynamics of molecules, ed. R.G. Woolley (Plenum Press, New York, 1980) p. 357. 116 A.J. MacDermott and G.E. Tranter, Croat. Chem. Acta 62 (1989) 165. [ 171 M. Quack, Angew. Chem. Intern. Ed. Engl. 28 (1989) 571. [ 181 L.D. Barron, in: New developments in molecular chirality, ed. P.G. Mezey (Rluwer, Dordrecht, 1991) p. 1. [ 191 L.D. Barron, Chem. Phys. Letters 79 ( 198 1) 392. [20] L.D. Barron, Mol. Phys. 43 (1981) 1395.
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[ 211 L.D. Barron, Molecular light scattering and optical activity (Cambridge Univ. Press, Cambridge, 1982). [22] P. Jungwirth, L. Skala and R. Zahradnfk, Chem. Phys. Letters 161 (1989) 502. [23] G. Ltiders, Ann. Phys. 2 (1957) 1. [24] R.F. Streater and A.S. Wightman, PCT, spin and statistics, and all that (Benjamin/Cummings, Menlo Park, 1964/ 1978). [25] J. Eades, R.J. Hughes and C. Zimmermann, Physics World July (1993) 44. [26] F. Wilczek, Fractional statistics and anyon superconductivity (World Scientific, Singapore, 1990).
[ 271 B.I. Halperin, J. March-Russell and F. Wilczek, Phys. Rev. B 40 (1989) 8726. [28] L.D. Barron, Chem. Phys. Letters 135 (1987) 1. [29] L.D. Barron, in: Proceedings, Conference on Chemical Evolution and the Origin of Life, International Centre for Theoretical Physics, Trieste, Italy, Oct. 1992, eds. C. Ponnamperuma and J. Chela-Flores (Deepak, Hampton), in press. [30] E.P. Wigner, Sci. Am. 213 (1965) 28. [31] L.R. Hunter, Science 252 (1991) 73. [ 321 A. Aharony, in: Modem developments in thermodynamics, ed. B. Gal-Or (Wiley, New York, 1973) p. 95.
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 22 1 ( 1994) 3 1l-3 16
CP violation and molecular physics L.D. Barron Chemistry Department, The University, Glasgow G12 8QQ, UK Received 29 April 1993; in final form 6 January 1994
Abstract The proof that a particle and its antiparticle have identical rest mass, which is based on the CPT invariance of the Hamiltonian, can be adapted to show that the CP enantiomers of a chiral molecule remain degenerate even in the presence of CP violation. This reinforces an earlier suggestion that the force responsible for CP violation exhibits false chirality with respect to CP enantiomorphism and leads naturally to the idea that the CP violation observed in the neutral K-meson system is analogous to a special type of chemical catalysis since it affects the rates of certain particle-antiparticle interconversion pathways (via a breakdown in microscopic reversibility) without affecting the initial and final particle energies and hence the equilibrium thermodynamics. CP violation of this type is unlikely to have any direct manifestations in molecular physics.
1. Introduction
The fundamental discrete symmetries of parity (P) , time reversal (T) and charge conjugation (C), and their violations in certain situations, are central to modern elementary particle physics (see e.g. refs. [ l31). In the realm of atomic and molecular physics, interest in symmetry violation has centred on parity violation induced by the weak neutral current interaction, manifestations of which are now observed routinely in polarized atomic spectroscopy experiments (see e.g. refs. [ 4,5 ] ). Since the combined CP symmetry is known to be violated in certain elementary particle processes [ 6 1, the possibility that CP violation might appear in atomic and molecular physics has also attracted some discussion [ 7,8] which has of necessity been speculative on account of the current incomplete understanding of this phenomenon [9-l 11. This Letter presents a new theorem conceming CP violation in chiral molecules, together with an analogy between CP violation in the neutral K-meson system and chemical catalysis, which emphasise
that CP violation falls within the conceptual framework of false chirality [ 12,13 1. These new insights provide a cornerstone for future discussions of CP violation in molecular physics.
2. True enantiomers Much has been made of parity violation in the special case of chiral molecules where the degeneracy of the mirror-image enantiomers is lifted (see e.g. refs. [ 4,14-l 8 ] ). Some years ago, I showed that the true enantiomer of a chiral molecule (i.e. the ‘opposite’ object with identical energy to the original) is the molecule with the opposite absolute configuration but composed of antiparticles, being generated from the original by the combined CP operation [ 19-22 ] ; this means that a chiral molecule is associated with two distinct pairs of true enantiomers (Fig. 1) . However, contrary to the impression given recently by Quack [ 81, I never intended that the term ‘enantiomers’ not be used for an ordinary optically active molecule and
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDi0009-2614(94)00253-M
312
L.D. Barron /Chemical Physics Letters 221(1994) 311-316 P
M
c,
h-l=
where O= CPT is the associated operator. Taking the matrix element of both sides between any two particle states labelled a and b, we have (b]H]a)=(blO-‘HSZ]a) =(~bl~~-‘Hnla)*=(~lH~la)*.
i
c--,
P
-m
M
Fig. 1. The two pairs of true (strictly degenerate) enantiomers (M, IP and M*, M) of a chiral molecule that are interconverted by CP.
its mirror image in ordinary stereochemistry; I was making the academic point that these are not strict opposites (I use the terminology ‘P-enantiomers’ and ‘CP-enantiomers’ when discussing these fundamental aspects [ 18 1, but not in ordinary stereochemistry). Quack then goes on to say that, since CP itself is known to be violated, we should allow in principle for the degeneracy of even CP enantiomers to be lifted by some CP violating interaction in molecules [ 8 1. My earlier papers also allowed for this possibility; but I now realize that, provided CPT invariance holds, the idea is wrong since, as proved below, CP enantiomers remain strictly degenerate even in the presence of CP violation.
3. CP enantiomers are always true The CPT theorem of relativistic quantum field theory states that, even if one or more of the operations C, P and T is violated, the combined operation of CPT is always conserved [ 23 1. This theorem can be proved in great generality with only mild assumptions [ 241. There are three important consequences [ 1,2,11]: the rest mass of a particle and its antiparticle are equal; the particle and antiparticle lifetimes are the same (even though decay rates for individual channels may not be equal); and the electromagnetic properties such as charge, magnetic moment, etc. are opposite. We are particularly interested in the mass, and hence energy, equality which is proved in the followingway [1,2,11]. Invariance of any Hamiltonian H under CPT is expressed by Q-‘HQ=H,
(1)
(2)
The second equality follows from the antiunitary nature of 8 associated with the time reversal operator contained within it (so that (8y/] @r) = (v/I 9) *). When applied to the state of a particle at rest, the operator g has the following effect:
Qla>=tla>,
(3)
where a labels the corresponding antiparticle state and
(4)
Taking a and b to denote the state of a single particle at rest, (4) becomes the real expectation value (a]H]a)
= (i%lHla)*= (alHI%)
.
(5)
Hence if i contains all the internal interactions which contribute to the particle mass, the mass equality of particle and antiparticle is proved. If the particle has spin, the spin angular momentum vector of the antiparticle will point in the opposite direction (because of T in a); but this feature is immaterial for a free particle because the condition of Lorentz invariance required for the derivation of the CPT theorem covers rotational symmetry so the original spin angular momentum direction can always be restored without affecting the energy. We now come to an important point. The standard derivation given above assumes that the P part of 52 does not alter the particle itself [ I] (although if the particle is not at rest its helicity will be reversed by P). But if a is now taken to denote the state of a stationary chiral molecule, we must allow for P to generate the mirror-image enantiomeric state denoted a”. Hence (3) must be generalized to
Qla>=TlaX>,
(6)
so that ( 5) becomes (alHla)=(aX]Hla”),
(7)
which proves the rest-mass, and hence energy, equiv-
L.D. Barron /Chemical PhysicsLetters221(1994) 31 l-316
alence of the CP enantiomers of a chiral molecule. Since only thefull CPT invariance of the Hamiltonian is invoked in the derivation, this result applies if any of C, P and T either separately or in combination are violated. Hence the CP enantiomers of a chiral mol-
ecule remain strictly degenerate even in the presence of CP violation. If the molecule has non-zero spin angular momentum associated with an odd number of electrons (or rotational angular momentum if in the gas phase) the result is not affected because the argument in the previous paragraph still applies. A similar extension of the proof of the lifetime equality of particle and antiparticle from CPT invariance [ 1,2,11] can be easily developed, if required, to take account of the fact that a chiral molecule is not in a stationary state of its Hamiltonian [ 17,18,21]. The above analysis shows that an energy difference between the CP enantiomers of a chiral molecule requires CPT violation. Hence any attempt to measure this energy difference would in fact constitute a search for both CP and CPT violation, not just CP violation as previously supposed [ 8 1. Although not feasible at present due to the current inaccessibility of antimolecules, such measurements are nonetheless interesting to contemplate in view of the recent proposal to use ultra high resolution spectroscopy to compare the lS-2s energy intervals in atomic hydrogen and antihydrogen to one part in 1O’* (which should soon be possible) as a test of CPT invariance to very high precision [ 25 1.
4. False chirality and CP violation The concept of false chirality was originally introduced to distinguish time-invariant P-enantiomorphism (‘true chirality’) from time-noninvariant P-enantiomorphism (‘false chirality’) [ 12,13 1. An example of the former is a conventional pair of mirror-image chiral molecules where two distinguishable enantiomers are interconverted by P but not by T. An example of the latter is a system of collinear electric and magnetic fields where two distinguishable enantiomers (the parallel and antiparallel arrangements) are interconverted not only by P but also by T so that it is PT invariant overall. Although not referred to as such, a version of false chirality also
313
arises in the anyon theory of high temperature superconductivity [ 26 ] where a spinning system in two dimensions breaks both P and T separately but again is invariant under the combined PT operation (parity has a different effect in two dimensions than in three since it becomes simply a reflection in only one axis t271). It has been pointed out previously that any force responsible for CP violation exhibits an analogue of false chirality with respect to CP enantiomorphism [ 18,28 1. This is because CP-violating forces always come in pairs that are interconverted by CP, with CPT invariance guaranteeing that the two distinct CP-enantiomeric forces are also interconverted by T. Violation of CP arises because only one of a CP-enantiomeric pair of forces is found in our universe. Having shown in section 3 that, if CPT invariance holds, no CP-violating force can lift the degeneracy of CP enantiomers (be they particle-antiparticle pairs or a chiral molecule and its mirror image made of antiparticles), the analogy is now satisfyingly complete since a falsely chiral P-enantiomorphous influence such as collinear electric and magnetic fields also cannot lift the degeneracy of conventional P-enantiomers such as a chiral molecule and its mirror image.
5. CP violation and chemical catalysis Violation of CP symmetry was first observed in 1964 in certain decay modes of the neutral K-meson [ 6 1. Although unequivocal, the effects are very small and to date have not been observed in any other system. On account of the CPT theorem, these experiments are also widely regarded as indirect observations of T violation (which has not yet been observed directly). One manifestation of CP violation is the following decay rate asymmetry of the long-lived neutral K-meson, the I&_[ 1,2,9- 111: d_ rate (KL+IC-e,+V~) x l 00648 rate (KL+x+e;O,) * .
As indicated, KL can decay into either positive pions A+ plus left helical electrons ep plus right helical antineutrinos &,;or into negative antipions A- plus right helical positrons e: plus left helical neutrinos vp. Since these two sets of decay products are intercon-
314
L.D. Barron I Chemical Physics Letters 221(1994) 31 l-316
verted by CP, the observed decay rate asymmetry is a manifestation of CP violation. If we naively incorporate these two decay processes into the following single ‘chemical equilibrium’ scheme involving the intermediate KL,
(9) we would say that (8) corresponds to an asymmetry in the rate constants for decay of KL into the two sets of CP-enantiomeric products, i.e. /q#@. This establishes a parallel with the absolute asymmetric synthesis which I have shown might be induced by a falsely chiral influence such as collinear electric and magnetic fields [ 18,281. As pictured in Fig. 2 for a unimolecular process in which an achiral molecule R generates a chiral molecule M or its enantiomer M +, the special feature is a breakdown in microscopic reversibility since the forward and backward potential energy profiles for reaction of a given enantiomer are different: this arises because the falsely chiral influence breaks P and T separately. However, the deeper principle of enantiomeric microscopic reversibility still holds because the influence preserves the combined PT invariance which is manifest here in the form of identical potential energy profiles for the forward and backward enantiomeric asymmetric synthesis reactions, i.e. for R-M” and M+R. Clearly a falsely chiral influence acts as a special type of catalyst in the chemical sense since it modi-
I
M'
M Reaction coordinate
Fig. 2. Potential energy profiles for enantiomeric reactions in the presence of a falsely chiral (time-noninvariant enantiomorphous) influence.
fies potential energy barriers to change relative rates of formation of enantiomeric products without affecting the relative energies of reactants and products (remember that a falsely chiral influence does not lift the degeneracy of P-enantiomeric chiral molecules) so that the position of the equilibrium is not changed [ 29 1. Since K,_and its two sets of decay products are the equivalents with respect to CP of R, M and MU with respect to P, we can conceptualize its decay rate asymmetry as arising from a breakdown in microscopic reversibility due to a time-noninvariant CP enantiomorphism in the forces of nature. The analogy is completed by the fact that in both cases the asymmetries cancel out when summed over all possible channels at true thermodynamic equilibrium [ 281. Since the CP-violating interaction responsible for the decay rate asymmetry of the KL does not lift the degeneracy of the two sets of CP-enantiomeric products, itsfunction is analogous to that of a special type of chemical catalyst.
6. CP violation in molecular physics
The long-lived and short-lived K-meson species KL and K, the decay modes of which show CP violation, are peculiar in that, being certain particle-antiparticle superposition states (which Wigner [ 301 has pointed out are analogues with respect to CP of the definite parity states 2 - ‘I2( vL ? VR) of a chiral molecule involving superpositions of the left- and righthanded P-enantiomeric states yL and @&,but slightly mixed with each other through CP violation), they constitute observable states bridging the worlds of matter and antimatter. As the equilibrium (9) suggests, these superposition states can be thought of as intermediates in particle-antiparticle transformation pathways. We might be tempted to deduce from this that CP violation and the associated breakdown in microscopic reversibility of the type observed in the neutral K-meson system could occur in principle in processes involving transformations between a chiral molecule made of matter and its currently inaccessible mirror image made of antimatter (i.e. molecular CP enantiomers), with an analogue of the K,_ or Ks involving molecule-antimolecule superposition states as an intermediate. Unfortunately, there are several fundamental impediments to this idea: for
L.D. Barron /Chemical Physics Letters 221(1994) 311-316
example, such molecule-antimolecule transformations would require a gross violation of the law of baryon number conservation which does not arise in the K-meson system because mesons have baryon number zero; and strange particles like K-mesons are probably required because specific strangeness quantum number changes are essential requirements of the current ‘superweak’ and ‘milliweak’ theories [ 9,101 of CP violation in these systems. Hence the type of CP violation observed in the Kmeson system seems unlikely to have any direct manifestations in molecular physics even if antimolecules were accessible. However, if CP and hence T violation could somehow infiltrate into the world of ordinary atoms and molecules it might, as recently suggested [ 8 1, be observed indirectly in the form of a breakdown in microscopic reversibility in molecular dynamics. And of course we can still attempt to mimic CP violation by using a falsely chiral influence (such as collinear electric and magnetic fields) to induce a breakdown in microscopic reversibility in certain processes involving ordinary chiral molecules. The necessity for particle-antiparticle processes which appears to prevent manifestations of CP violation appearing directly in atomic and molecular physics can be avoided if T is violated directly. A central example of this is a permanent electric dipole moment in a definite parity atomic or molecular state which requires simultaneous violation of both T and P and which, despite much experimental effort, has not yet been detected (see e.g. refs. [4,9-l 1,16,3 1] ). Apart from involving only ordinary matter, this situation appears to be fundamentally different from that of the neutral K-meson system since a stationary state, rather than a process, is being studied. Finally it is not widely appreciated that, as proved by Aharony [ 321, Boltzmann’s H-theorem and hence also the second law depend on unitarity rather than microscopic reversibility and so still hold even if the particles making up a system are subject to forces that violate T-invariance (although the time dependence of the H-function could be modified during the approach to equilibrium). This serves to reinforce Sachs view that there is no direct connection between the ‘arrow of time’ (the irreversibility of macroscopic phenomena) and the physics of time reversal [ 111. It also casts light on the speculation that, while it is not necessary to invoke T violation to ex-
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plain irreversibility in statistical mechanical molecular chaos and the macroscopic world generally, an explanation based on T violation cannot be ruled out [8]: it appears that, at most, T violation can only constitute a small perturbation to the rate at which equilibrium is attained.
Acknowledgement
I thank Professor M. Quack for sending me a copy of ref. [ 81 in advance of publication and for much stimulating correspondence, and Dr. C.D. Froggatt, Dr. L. Hecht and Dr. D.G. Sutherland for discussion.
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