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Hybrid Dirac Fields Physics Letters B 567 (2003) 179–183 www.elsevier.com/locate/npe Hybrid Dirac fields O.W. Greenberg Center for Theoretical Physics, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA Received 23 May 2003; accepted 25 June 2003 Editor: H. Georgi Abstract Hybrid Dirac fields are fields that are general superpositions of the annihilation and creation parts of four Dirac spin 1/2 ± fields, ψ( )(x;±m), whose annihilation and creation parts obey the Dirac equation with mass m and mass −m.Wediscuss a specific case of such fields, which has been called “homeotic”. We show for this case, as is true in general for hybrid Dirac fields (except the ordinary fields whose annihilation and creation parts both obey one or the other Dirac equation), that (1) any interacting theory violates both Lorentz covariance and causality, (2) the discrete transformations C,andCPT map the pair ¯ ψh(x) and ψh(x) into fields that are not linear combinations of this pair, and (3) the chiral projections of ψh(x) are sums of the usual Dirac fields with masses m and −m; on these chiral projections C,andCPT are defined in the usual way, their interactions do not violate CPT , and interactions of chiral projections are Lorentz covariant and causal. In short, the main claims concerning “homeotic” fields are incorrect. 2003 Published by Elsevier B.V. 1. Introduction fields. This type of field, with a specific choice given below, was considered in a recent paper by Barenboim Since a spin 1/2 field on which parity is defined can and Lykken [1] (BL) in connection with a proposed obey either of two Dirac equations model of CPT violation for neutrinos. They called their field a “homeotic” field. The purpose of this Let- (i/∂ − m)ψ(x; m) = 0 (1) ter is to study the properties of this type of field which, or for reasons stated above, I prefer to call a “hybrid (i/∂ + m)ψ(x;−m) = 0 (2) Dirac field”. BL proposed their “homeotic” field as a counter-example to my general theorem [2] that inter- and since the positive (annihilation) and negative (cre- acting fields that violate CPT symmetry necessarily ation) frequencies of a free field can be separated in a violate Lorentz covariance. I will show below that the Lorentz covariant way, one can consider a family of BL example does not violate CPT and that their inter- free “hybrid Dirac fields” which, with suitable nor- acting “homeotic” field violates both Lorentz covari- malizations, are linear combinations of the annihila- ance and causality. tion and creation parts of mass m and mass −m Dirac 2. The simplest representation of the BL field is E-mail address: [email protected] (+) (−) (O.W. Greenberg). ψh(x) = ψ (x; m) + ψ (x;−m), (3) 0370-2693/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.physletb.2003.06.046 180 O.W. Greenberg / Physics Letters B 567 (2003) 179–183 where to be explicit and to establish notation, of the normalizations of the u and v spinors for the −m case may seem strange; however these normaliza- ψ(+)(x;±m) tions are consistent with the anticommutation relation 1 d3p for the −m field, with the positivity of energy (note = b(p,s)u(p,s;±m) 3/2 that the u†u normalization is positive for both the m (2π) 2Ep ± s and −m spinors) and with the relation between m and × − · exp( ip x), (4) −m fields via γ 5 given below. If I had started with − ψ( )(x;±m) the −m field rather than with the m field, I could have 3 chosen gammas so that the −muspinors would have 1 d p † = d (p, s)v(p, s;±m) positive normalization. (2π)3/2 2E p ±s (Other linear combinations of the positive and × exp(ip · x), (5) negative frequency parts of Dirac fields of mass m and − − ψ¯ ( )(x;±m) mass m also fall into the category of hybrid Dirac fields as do the usual Dirac fields, but in this Letter 1 d3p = b†(p, s)u(p,s¯ ;±m) I discuss only the example given above.) I follow the 3/2 (2π) 2Ep 0 = = ±s universal convention, [3,4] and [5], that the p Ep × exp(ip · x), (6) m2 + p2 in spinors is always the positive energy; ψ¯ (+)(x;±m) unfortunately BL violate this convention. 3 To see that my representation of ψh(x) is identical = 1 d p ¯ ;± with the BL field, calculate the anticommutator 3/2 d(p,s)v(p,s m) (2π) 2Ep ± s (+) [ψ (x), ψ¯ (y)]+ = (i/∂ + m) ∆ (x − y) × exp(−ip · x). (7) hα hβ x αβ (−) − (i/∂x − m)αβ ∆ (x − y), + ¯ (−) Note the difference between ψ( ) = ψ and (11) ¯ (+) (+) ψ , etc. To separate ψh(x) into ψ (x; m) (−) where and ψ (x;−m),use[(±i/∂x + m)/2m]ψh(x) = (±) ;± 3 ψ (x m); b(p,s), etc., can then be calculated in (+) = 1 d p − · ∆ (x) 3 exp( ip x), the usual way. The annihilation and creation operators (2π) 2Ep are normalized covariantly, − + ∆( )(x) = ∆( )(−x). (12) [ † ] = − + b(p,s),b (q, t) + 2Epδ(p q)δst, (8) This agrees with (2.14) of BL, (their D(x) = ∆( )(x)), etc., and the spinors obey except that their Dirac indices are transposed; how- ever, this differs from the Dirac result by the sign of (/p ∓ m)u(p, s;±m) = 0, (9) the second term proportional to m, not that of the first (/p ± m)v(p, s;±m) = 0, (10) term, as stated by BL. It is instructive to rewrite this result as with normalizations u(p,¯ s;±m)u(p, t;±m) = [ ¯ ]+ ±2mδst and v(p,s¯ ;±m)v(p, t;±m) =∓2mδst. ψhα(x), ψhβ (y) (1) I chose these normalizations so that going from = (i/∂x )αβ i∆(x − y)+ mδαβ ∆ (x − y), (13) ψ(±)(x;±m) to ψ(±)(x;∓m) is accomplished by = (+) − (−) (1) = (+) + (−) changing the sign of m. (This would not be the case where i∆(x) ∆ ∆ and ∆ ∆ ∆ , since this separates the local and nonlocal terms. if I had chosen uu¯ = δst, for example.) I use the same = (+) ; − (−) ; non-chiral Dirac matrices of [3] and [4] and the same In addition, i/∂ψh(x) m(ψ (x m) ψ (x − creation and annihilation operators for both the m and m)); it is straightforward to check that −m fields. With this choice of gamma matrices, the + − m ψ( )(x; m) − ψ( )(x;−m) muspinors have large upper two components and the − −muspinors have large lower two components, and = im 1 P dt ψh(t , x) (14) vice versa for the v spinors. The reversal of the signs π x0 − t O.W. Greenberg / Physics Letters B 567 (2003) 179–183 181 (provided the integral is defined suitably). Thus the the Lorentz covariance of the time-ordering in the present ψh obeys the equation of motion (2.7) of BL. Dyson series, and the Lorentz invariance of their In agreement with my result [2] and with BL, S-matrix are all incorrect and their interacting theory free, or generalized free, hybrid Dirac fields transform is neither Lorentz covariant nor causal. covariantly. This is connected with the existence of In contrast to my disagreement with the assertions θ(±p0) which exists for timelike momenta, but does concerning equal time commutation relations for the not exist for spacelike momenta. I also agree with BL currents and the Hamiltonian density, I agree with BL that the propagator of their field ψh(x) is not causal; it about the equal time commutation relations for the is also not covariant. chiral projections of the hybrid fields. The agreement From Eq. (1), (2), γ 5ψ(x;±m) obeys the Dirac here is because the chiral projections of the hybrid equation for ψ(x;∓m),soitisusefultoconsiderthe fields are sums of the usual Dirac fields with masses discrete transformation m and −m. These chiral projections are not hybrid (or homeotic) fields at all! M ; M† = ;− = 5 ; ψ(x m) ψ(x m) γ ψ(x m), (15) I agree with BL that parity and time reversal Mψ(x;−m)M† = ψ(x; m) = γ 5ψ(x;−m). (16) are realized in the usual way. BL assert that charge conjugation is realized in a different way than usual. − This shows that the m and m spinors are related This is incorrect. This assertion seems to be based on 5 by γ . the tacit assumption that C and CPT map the pair The calculation of the BL current is particularly ¯ ψh(x) and ψh(x) onto themselves. Unexpectedly, this simple for the space components. The result is is not the case. The discrete transformations C and i j CPT map the terms in ψh(x) into terms that are not J (x), J (y) − ¯ ¯ present in ψh(x) and map the terms in ψh(x) into ¯ i j = ψh(x)γ i/∂x i∆(x − y)γ ψh(y) terms that are not present in ψh(x). ¯ j i Although it is well known, I emphasize that the re- − ψh(y)γ i/∂y i∆(y − x)γ ψh(x) quirement that charge conjugation changes the sign + m −δij [ψ¯ (x)ψ (y) − ψ¯ (y)ψ (x)] h h h h of the field and the requirement that charge conjuga- ¯ ij ¯ ij − i[ψh(x)σ ψh(y) + ψh(y)σ ψh(x)] tion interchanges particle and antiparticle are equiva- lent [6–8]. Here is the standard argument concerning × ∆(1)(x − y). (17) charge conjugation [4].
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