aa r X i v : . [ h e p - ph ] J un Modern Physics Letters A, ❢ c World Scientific Publishing Company Causal Propagation of Spin-Cascades
L. M. RICO, M. KIRCHBACH ∗ Instituto de F´ısica, Universidad Aut. de San Luis Potos´ı,Av. Manuel Nava 6, Zona Universitaria,San Luis Potos´ı, SLP 78290, M´exico
Received (received date)Revised (revised date)
Abstract
We gauge the direct product of the Proca with the Dirac equation that de-scribes the coupling to the electromagnetic field of the spin-cascade (1/2,3/2)residing in the four–vector spinor ψ µ and analyze propagation of its wavefronts in terms of the Courant-Hilbert criteria. We show that the differentialequation under consideration is unconditionally hyperbolic and the propaga-tion of its wave fronts unconditionally causal. In this way we proof that theirreducible spin-cascade embedded within ψ µ is free from the Velo-Zwanzigerproblem that plagues the Rarita-Schwinger description of spin-3/2. The proofextends also to the direct product of two Proca equations and implies causalpropagation of the spin-cascade (0,1,2) within an electromagnetic environ-ment. Keywords : High-spins; causal propagation.
1. Introduction
The consistent description of high-spins within a covariant framework is a longstanding problem in particle physics. The commonly used description of fermionswith spin higher than 1 / J = ( K + 1 /
2) with K integer as the highest spin in the totallysymmetric rank– K tensor spinor ψ µ µ ...µ K and describe it by means of the Diracequation, ( p − m ) ψ µ µ ...µ K = 0 , (1) ∗ E-mail: mariana@ifisica.uaslp.mx 1
L. M. Rico, M. Kirchbach as supplemented by the two auxiliary conditions p µ ψ µ µ ...µ K = 0 , (2) γ µ ψ µ µ ...µ K = 0 , (3)with p µ being the four momentum. In so doing one restricts the degrees of freedomto 2(2 J + 1) which are then associated with spin- J particles and antiparticles. Thetensor-spinor representation spaces reside in the direct products of K four-vectorcopies with the Dirac spinor ψ = (1 / , ⊕ (0 , /
2) according to ψ µ µ ...µ K = Sym(1 / , / ⊗ (1 / , / ⊗ ... ⊗ (1 / , / K ⊗ [(1 / , ⊕ (0 , / . (4)They consist in their rest frames of K parity doublets with spins ranging from 1 / ± to ( K − / ± while the highest spin J = K + 1 / ψ µ µ ...µ K rest frame −→ + , − ; 32 + , − ; ... ; (cid:18) K − (cid:19) + , (cid:18) K − (cid:19) − ; (cid:18) K + 12 (cid:19) π . (5)Here the parity of the highest spin is ( − K for tensors, and ( − K +1 for pseudo-tensors. Equation (5) illustrates the meaning of ψ µ µ ...µ K as spin-cascades. Thetensor-spinor of lowest rank is the four-vector spinor, ψ µ . It is used in the descriptionof spin-3/2, in which case one encounters the shortest spin-cascade ψ µ rest frame −→ + , − ; 32 − . (6)Here we choose the polar four-vector spinor for concreteness. Applied to ψ µ ,Eqs. (1)–(3) reduce the 16 degrees of freedom to 8 and associate them with particlesand anti-particles of spin-3/2 at rest. The first auxiliary condition in Eq. (2) ex-cludes the spin-0 + component of the four-vector and thereby the spin-1 / + part of ψ µ , while the second auxiliary condition in Eq. (3) excludes its parity counterpartspin-1 / − . The tensor-spinor framework can be given a Lagrangian formulationupon establishing the most general form of a Lagrangian that leads to the abovethree equations. Such a Lagrangian for the four-vector–spinor ψ µ can be found in3 and reads L ( A ) = ¯ ψ µ [ p α Γ αµ ν − mg µν ] ψ ν , (7)where p α Γ αµ ν ( A ) ψ ν = pψ µ + B ( A ) γ µ p γ · ψ + A ( γ µ p · ψ + p µ γ · ψ ) + C ( A ) mγ µ γ · ψ ,A = 12 , B ( A ) ≡ A + A + 12 , C ( A ) = 3 A + 3 A + 1 . (8)The wave equation following from the above Lagrangian is obtained as( p − m ) ψ µ + A ( γ µ p · ψ + p µ γ · ψ ) + B ( A ) ( γ µ p γ · ψ ) + C ( A ) mγ µ γ · ψ = 0 , (9) ausal Propagation of Spin-Cascades which for A = − iε µνβα γ γ β p α − mg µν + mγ µ γ ν ) ψ ν = 0 . (10)Equations of this type are equivalent to12 m ( p + m ) ψ µ = ψ µ , (11)( − g µν + 1 m p µ p ν ) ψ ν = − ψ µ , (12) γ µ ψ µ = 0 , (13)known as the Rarita-Schwinger (RS) framework.2 Notice that for the sake of con-venience of the point we are going to make in the next section, we here wrote therespective Dirac and Proca equations (11), and (12) in terms of covariant projec-tors picking up spin-1/2 + and spin-1 − states, respectively. Spin 3/2 + needs an axialfour vector. Equation (11)–(13) suffer several inconsistencies one of them being thenon-causal propagation of the classical wave fronts of its solutions, a result dueto Ref.4 and known as the Velo-Zwanziger problem. Furthermore, the inverse ofEq. (9) does not relate to the spin 3/2 projector alone but is a more complicatedcombination of various projectors 5. This inconvenience can affect the quantizationprocedure which is also known to suffer inconsistencies 6.We here made the case that the Velo-Zwanziger problem is not inherentto the ψ µ representation space by itself but rather to its descriptionwithin the Rarita-Schwinger framework. To be specific, we shall showthat the propagation of the spin cascade (1 / , /
2) when described interms of the direct product of the Proca and Dirac equations does notsuffer the Velo-Zwanziger problem but propagates strictly causally. Inaddition, one gains coincidence between the nominator in the propagator(the latter being the inverse of the wave equation) and the projector builtfrom the states.Spin-cascade propagations are of interest in the physics of baryon resonances whereaccording to Ref. 7 (and references therein) one observes well pronounced mass–and parity degeneracies patterned after the tensor-spinors of rank-1, 3, and 5, re-spectively, but also possibly for the physics of the gravitino and the graviton, anidea that has been put forward in Refs. 8,9. The idea of using spin-cascades asgauge fields in unified theories has been pioneered by Kruglov and collaborators(see Ref. 10 and references there in).The paper is organized as follows. In the next section we briefly review theessentials of the Velo-Zwanziger problem of acuasal spin-3/2 propagation in thelight of the Currant-Hilbert criteria. Section III is devoted to the propagationproperties of the spin-cascades (1/2,3/2) and (0,1,2). The paper ends with a briefsummary.
L. M. Rico, M. Kirchbach
2. The Velo-Zwanziger problem of high-spin propagation
The non-causal propagation of spin-3/2 within the gauged Rarita-Schwingerframework has first been addressed in the work of Giorgio Velo and Daniel Zwanziger4.For the sake of self sufficiency of the presentation we here highlight it in brief. Themain point of Ref. 4 is that Eq. (10) provided by the Lagrangian (8) is not a genuinefirst order equation of motion because it does not contain any time derivative of ψ at all.This defect shows up in the (i) complete cancellation of all ∂ ψ terms in Eq. (10)for any µ , (ii) complete cancellation of all the ∂ ψ α terms for µ = 0, in which caseone finds instead of a wave equation the constraint[ p + ( p · γ − m ) γ ] · ψ = 0 , (14)(iii) absence of ψ in Eq. (14) that leaves the time-component of the Rarita-Schwinger field undetermined. The above deficits are caused by the constraintshidden in the wave equation and could be tolerated only if remediable upon gaug-ing. Velo and Zwanziger gauge Eq. (10) in Ref. 4 in replacing p µ by π µ = p µ + eA µ ,and succeed in constructing a genuine equation. Their remedy procedure beginswith first contracting the gauged equation successively by γ µ and π µ and obtainingthe covariant gauged constraints as γ · ψ = − iem γ γ · e F · ψ , (15) π · ψ = − ( γ · π + 32 m ) 23 iem γ γ · e F · ψ , (16)and ends with substituting Eqs. (15,16) back into the gauged Eq. (10). The resultingnew wave equation,( π − m ) ψ µ + ( π µ + m γ µ ) 23 iem γ γ · e F · ψ = 0 , (17)is now a genuine one because it can be shown to determine both ψ and the timederivatives of ψ µ for any given µ .The final goal is to test hyperbolicity and causality of Eq. (17) by means of theCourant-Hilbert criterion 11 which requires the so called characteristic determinantof the matrix containing the highest derivatives when replaced by n µ , i.e. by thenormals to the characteristic surfaces, to vanish only for real n . The Courant-Hilbert criterion is applied in fact not directly to Eq. (17) but to its Hermitian formas obtained by using repeatedly Eqs. (15) and (16):( γ · π − m ) ψ µ + ( π µ + 12 mγ µ ) 2 ie m γ γ · e F · ψ + 2 ie m e F µ · γγ ( π + 12 mγ ) · ψ + 2 ie m e F µ · γγ ( γ · π + 2 m ) 2 ie m γ γ · e F · ψ = 0 (18) ausal Propagation of Spin-Cascades The last equation represents now a system of partial differential equations whichwould describe wave propagation phenomena provided it were hyperbolic, somethingthat can be tested by exploiting the Courant-Hilbert criterion. For this purpose itis sufficient to compute the normals n µ to the characteristic surfaces. To find thenormals to the characteristic surfaces passing through each point we replace i∂ µ by n µ in the highest derivatives and calculate the determinant D ( n ) of the resultingcoefficient matrix, which is called the characteristic determinant. The equation ofmotion Eq. (18) will be hyperbolic if the solutions n to D ( n ) = 0 are real forany n µ = ( n , n ). By means of this prescription we are left with the followingcharacteristic determinant for the Rarita-Schwinger framework, D ( n ) = (cid:12)(cid:12)(cid:12)(cid:12) γ · n g νµ + 2 ie m n µ γ γ · e F ν + 2 ie m e F µ · γγ n ν + (cid:18) ie m (cid:19) e F µ · γγ ( γ · n ) γ γ · e F ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (19)The covariant form of the latter is found to be, D ( n ) = ( n ) " n + (cid:18) e m (cid:19) ( e F · n ) = 0 . (20)Equation (20) has the following four positive and four negative roots n = ±√ n . (21)Eight more roots are found from n = ± r ( n − k ( B · n ) )1 − k B = ± p n (1 − k B cos λ )(1 − k B )1 − k B , (22)respectively, where B stands for the magnetic field, k = e m , and λ is the anglebetween B and n . Introducing the notion of the ”weak-field limit” to refer to thesituation in which there exists, for each space-time point, a Lorentz frame such thatthe inequality, k B ≤ , (23)is satisfied, we see that the roots given by Eq. (22) are real for any given n = ( n , n ),which establishes hyperbolicity.In the strong-field limit when the inequality (23) is no longer satisfied, Eq. (18)ceases to be hyperbolic and is not suitable for the description of wave phenomenaat all.Moreover, in recalling that the maximum velocity of the signal propagation isthe slope of the characteristic surfaces, one immediately realizes that the character-istic surfaces determined by Eq. (20) are not all tangent to the light cone. Stateddifferently, one finds space-like characteristic surfaces passing through all points L. M. Rico, M. Kirchbach where F µν is non-vanishing. Consequently, such signals are propagated at veloc-ities greater than the speed of light. To see this it is just a matter of realizingthat there are time-like normals n µ which satisfy the characteristic equation (20).Thus, Eq. (18) has characteristic surfaces which propagate non-causally. In fact,for n µ = (1 , , ,
0) Eq. (20) takes the form of equality,1 = k B , (24)and whenever F µν = 0, there exists a Lorentz frame where the latter equation holdsvalid. The above considerations show that causal propagation requires uncondition-ally hyperbolic wave equations.So far only few authors have proposed solutions to the Velo-Zwanziger problemof the linear Rarita-Schwinger Lagrangian 12.13 In Ref. 12 Ra˜ n ada and Sierraelaborate a method which is equivalent to the on-shell Rarita-Schwinger frameworkbut in their gauged formalism the two auxiliary conditions have been replaced byone differential equation. In effect, the number of degrees of freedom increases fromeight to twelve and the theory in Ref. 12 describes spin-1/2 and spin-3/2 particles ofdifferent masses. In applying the Courant-Hilbert criterion to their wave equation,the authors of Ref. 12 prove that the wave fronts of the solutions of their equationindeed do propagate causally. More recently, Kruglov suggested two different waveequations, the first of which is of second order and non-local while the second islocal and linear. 13 None of the above equations suffers the pathology of non-causalpropagation of spin-3/2. The non-local equation has twelve degrees of freedom ofequal masses which are associated with spin-3/2 and spin-1/2. The local equationuses a twenty dimensional space to embed the four-vector spinor which containsnext to spin-3/2 also the two copies of spin-1/2 of opposite parities. Within thelatter scenario the three different spin sectors of ψ µ appear characterized by threedifferent masses. Apparently, the mass-splittings between the spins is the price tobe paid for the causal propagation of the solutions to linear wave equations. Finally,single spins of the type ( s, ⊕ (0 , s ) have been shown to propagate always causally,results due to Hurley 14, and more recently to Ahluwalia and Ernst.15 On theone side, Hurley’s focus is on the manifestly hyperbolic nature of the generalizedFeynman–Gell-Mann equations for ( s, ⊕ (0 , s ),( π − m )Ψ ( s, ⊕ (0 ,s ) = e s S µν F µν Ψ ( s, ⊕ (0 ,s ) , (25)which are of second order in the momenta, and obviously manifestly hyperbolic. Thesolutions of the generalized Feynman-Gell-Mann equations therefore propagate un-conditionally causally. Ahluwalia and Ernst construct the ( s, ⊕ (0 , s ) propagationfrom the different perspective of the representation space and obtain it as third orderin the momenta. Their causality proof is based on the correct energy–momentumdispersion relations. Despite their merits, the ( s, ⊕ (0 , s ) representations are notas popular because they are difficult to couple to the pion-nucleon or photon-nucleon ausal Propagation of Spin-Cascades system due to dimensionality mismatch, a reason that still represents an obstacleto their application in phenomenology.In the next section we (i) consider as a new option a local but third order single-mass wave equation describing the (1 / , /
2) cascade residing in ψ µ and, (ii) deliverthe proof of the causal propagation of its wave fronts within an electromagneticenvironment. The major appeal of the latter option is that it matches well withthe observed mass degeneracy of spin-1/2 − and 3/2 − baryons such like, say, the S (1535) and D (1520) resonances 7 etc. From that perspective it is desirable tohave a spin-cascade equation that is characterized by a single mass.
3. Cascade–spin description and causal propagation
In order to find the wave equation for the spin-cascade of interest it is quiteinstructive to go back to Eq. (4) and to recall the wave equation for the four-vectorspinor. Although one encounters various equations for the (1 / , /
2) representationspace in the literature, 16,17, 18,19 all they reflect different facets of the followingequation and related auxiliary condition: (cid:2) ( p − m )[ g ] νµ − p µ p ν (cid:3) A µ = 0 , (26) p µ A µ = 0 . (27)The latter equations imply that the four degrees of freedom of the (0 + , − ) spincascade in (1 / , /
2) have been reduced to three and are associated with spin-1 atrest. Now we consider the direct product of Eq. (27) with the Dirac equation( p − m ) ψ = 0 , (28)which will describe accordingly the spin-cascade (1 / − , / − ), (cid:2) ( p − m )[ g ] νµ − p µ p ν (cid:3) ⊗ ( p − m ) ψ µ = 0 ,ψ µ = A µ ⊗ ψ . (29)The inverse of the latter equation has the advantage to provide a propagator,Π (1 / , / ( p ) = 12 m (cid:0) − g µν + m p µ p ν (cid:1) ( p + m ) p − m + iǫ , (30)which is consistent with the Proca and Dirac projectors in Eqs. (12) and (11),respectively.Upon gauging, Eq. (26) becomes (cid:2) ( π − m )[ g ] νµ − π µ π ν (cid:3) ( π − m ) ψ µ = 0 , (31)where we dropped the ⊗ sign for simplicity. If now one calculates the characteristicdeterminant | Γ νµ | of the matrix that has as elements all the highest derivative termswith the derivatives being replaced by n µ , one finds it to be zero. This means that L. M. Rico, M. Kirchbach the latter equation has constraints built in which prevent it from being a genuinesystem of differential equations. This deficit is removed upon finding the gaugedauxiliary conditions and substituting them back into the leading equation.To do so we contract Eq. (31) by π ν with the result, (cid:2) π µ ( π − m ) − ( π µ π ν + ieF νµ ) π ν (cid:3) ( π − m ) ψ µ = 0 (cid:2) m π µ + ieF νµ π ν (cid:3) ( π − m ) ψ µ = 0 , (32) m π µ ( π − m ) ψ µ = ieF µλ π λ ( π − m ) ψ µ . In order to incorporate the gauged auxiliary condition in Eqs. (32) into the waveequation we commute the canonical momenta to obtain π µ π ν in Eq. (31), (cid:2) ( π − m )[ g ] νµ − ( π ν π µ + ieF νµ ) (cid:3) ( π − m ) ψ µ = 0 , (cid:8) ( π − m )[ g ] νµ ( π − m ) − π ν π µ ( π − m ) + ieF νµ ( π − m ) (cid:9) ψ µ = 0 , (cid:26) ( π − m )[ g ] νµ − iem π ν F µλ π λ + ieF νµ (cid:27) ( π − m ) ψ µ = 0 . (33)The following terms contribute to the characteristic determinant | Γ νµ | : π π [ g ] νµ + iem π ν F λµ π λ π → Γ νµ = n n [ g ] νµ + n ν iem F λµ n λ n . (34)The determinant to be calculated can be cast into the form | Γ νµ | = (cid:12)(cid:12)(cid:12)(cid:12) n n [ g ] νµ + n ν iem F λµ n λ n (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) n [ g ] νµ + n ν iem F λµ n λ (cid:19) ⊗ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) . (35)The latter equation is no more but the direct product of [Γ νµ ] Proca with [Γ νµ ] Dirac .We now can use the general theorem on the determinant of the direct (Kronecker)product of two matrices, here denoted by A , and B , of dimensionality n and q ,respectively, which tells one that | A ⊗ B | = | A | q | B | n . (36)In applying this formula to the direct product of[Γ νµ ] Proca = ( n ) , and [Γ νµ ] Dirac = ( n ) , (37)under consideration, we here find[Γ νµ ] Proca ⊗ Dirac = ( n ) , (38)and arrive at the satisfactory result on its unconditional hyperbolicity.This result can be independently confirmed by using appropriate routines insymbolic mathematical codes such as Maple in the calculation of the | Γ νµ | determi-nant as | Γ νµ | = | n n [ g ] νµ + n ν iem F λµ n λ n | = ( n ) . (39) ausal Propagation of Spin-Cascades The conclusion is that spin-cascades are allowed to propagate causally within theelectromagnetic environment. Same is valid for the Kronecker products of arbitrary(finite) number of Proca equations on the basis of Eq. (36). An interesting caseis the one of two Proca equations where one encounter the spin-cascade (0 , , ψ µ µ sector characterized by a single spin-2 at rest, a result due to.19
4. Summary
To summarize, we here found that Kronecker products of a finite number ofProca equations with or without the Dirac equation allow for causal propagation ofthe wave fronts of the associated solutions. Such solutions, in carrying factorizedLorentz and spinor indices are naturally coupled to the pion-nucleon, or photon-nucleon systems and describe inseparable spin-cascades. The corresponding waveequations provide propagators that are consistent with the projectors onto the statesand we expect this feature to facilitate the quantization procedure. Spin cascadessuch like (1/2,3/2) and (0,1,2) may be of interest both to particle spectroscopy andgravity.
5. Acknowledgments
We appreciate insightful discussions with Mauro Napsuciale on causality in thelight of the Courant-Hilbert criterion.Work supported by Consejo Nacional de Ciencia y Tecnolog´ıa (CONACyT, Mex-ico) under grant number C01-39820.
References
1. M. Fierz, W. Pauli, Proc. Roy. Soc. (London)
A173 , 211 (1939).2. W. Rarita, J. Schwinger, Phys. Rev. , 61 (1941).3. P. A. Moldauer, K.M. Case, Phys Rev. , 279 (1956).4. G. Velo, D. Zwanziger, Phys. Rev. , 1337 (1969).5. J. Weda, Spin-3/2 particles and consistent πN ∆ and γN ∆ -couplings , Ph. D. the-sis, KVI, University of Croningen, July, 1999.6. K. Johnson, E. C. Sudarshan, Annals of Physics , 126 (1961).7. M. Kirchbach, M. Moshinsky, Yu. F. Smirnov, Phys. Rev. D64 , 114005 (2001).8. M. Kirchbach, D. V. Ahluwalia, Phys. Lett.
B529 , 124 (2002).9. D. V. Ahluwalia, N. Dadhich, M. Kirchbach, Int. J. Mod. Phys. D , 1621 (2002).10. S. I. Kruglov, Symmetry and Electromagnetic Interactions of Fields with Multi-Spin (Nova Science Publishers, Huntington, N.Y. 2001) p. 216.11. R. Courant, D. Hilbert,
Methods of Mathematical Physics (Wiley Inter-science, NewYork, 1962).12. A. F. Ra ˜ n ada, G. Sierra, Phys. Rev. D22 , 2416 (1980).13. S. I. Kruglov, Int. J. Mod. Phys.
A21 , 1143 (2006).14. W. J. Hurley, Phys. Rev. D
4, 3605 (1971).15. D. V. Ahluwalia, D. J. Ernst, Int. J. Mod. Phys. E2 , 397 (1996). L. M. Rico, M. Kirchbach
16. L. H. Ryder,
Quantum Field Theory (Cambridge University Press, 1985)17. D. V. Ahluwalia, M. Kirchbach, Mod. Phys. Lett.
A16 , 1377 (2001).18. M. Napsuciale, C. A. Vaquera-Araujo,
Equations of motion as projectors and thegyromagnetic factor g(s)=1/s from first principles , hep-ph/0310106.19. G. Velo, D. Zwanziger, Phys. Rev.188