Beginnings of the Helicity Basis in the (S, 0) (0, S) Representations of the Lorentz Group

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1 Beginnings of the Helicity Basis in the (S, 0 (0, S Reresentations of the Lorentz Grou Valeriy V. Dvoeglazov UAF, Universidad Autónoma de Zacatecas, México valeri@fisica.uaz.edu.mx Abstract We write solutions of relativistic quantum equations exlicitly in the helicity basis for S = / and S =. We resent the analyses of relations between Dirac-like and Majorana-like field oerators. Several interesting features of bradyonic and tachyonic solutions are resented. Introduction. In Refs. [, ] we considered the rocedure of construction of the field oerators ab initio (including for neutral articles. The Bogoliubov-Shirkov method has been used. In the resent article we investigate the helicity h = / and h = cases in the helicity basis. We look for relations between the Dirac-like field oerator and the Majorana-like field oerator. In the first art we refer to the reviously found contradiction in the construction of the Majorana-like field oerator for sin /. In the nd art we analize the Majorana-like field oerator in the (, 0 (0, reresentation. It seems that the calculations in the helicity basis only give mathematically and hysically reasonable results.

2 Helicity Basis in the (/, 0 + (0, / Reresentation. The Dirac equation is: [iγ µ µ mc/ h]ψ(x = 0, µ = 0,,, 3. ( The γ µ are the Clifford algebra matrices γ µ γ ν + γ ν γ µ = g µν, ( g µν is the metric tensor. Usually, everybody uses the definition of the field oerator (in Ref. [3] in the seudo-euclidean metrics as given ab initio. After actions of the Dirac oerator on ex( i µ x µ the 4-sinors ( u and v satisfy the momentum-sace equations: (ˆ mu h ( = 0 and (ˆ+mv h ( = 0, resectively; the h is the olarization index. It is easy to rove from the characteristic equations Det(ˆ m = ( 0 m = 0 that the solutions should satisfy the energy-momentum relations 0 = ±E = ± + m for both u and v solutions. However, the general scheme of construction of the field oerator has been resented in [4]. In the case of the (/, 0 (0, / reresentation we have: Ψ(x = d 4 δ( m e i x Ψ( = (π 3 = d 4 δ( (π 3 0 Ee i x u h ( 0, a h ( 0, = h d 4 = [δ( (π 3 0 E + δ( 0 + E ][θ( 0 + θ( 0 ]e i x E u h (a h ( = d 4 [δ( h (π 3 0 E + δ( 0 + E ] (3 E h [ θ( 0 u h (a h (e i x + θ( 0 u h ( a h ( e +i x] = d 3 = θ( (π 3 0 [ u h (a h ( 0 =E E e i(et x + h + u h ( a h ( 0 =E e +i(et x] During the calculations we had to reresent = θ( 0 + θ( 0 above in order to get ositive- and negative-frequency arts. Moreover, we did not

3 yet assumed, which equation this field oerator (namely, the u sinor satisfies, with negative- or ositive- mass and/or 0 = ±E. In general we should transform u h ( to the v h (. The rocedure is given below [, ]. The exlicit forms of the 4-sinors are very well known in the sinorial basis: u σ ( = N + σ m(e + m ( [E + m + σ ]φ σ (0, v [E + m σ ]χ σ (0 σ ( = γ 5 u σ (, (4 ( where φ (0 = χ (0 = and φ 0 (0 = χ (0 = ( 0. The transformation to the standard basis is roduced with the (γ 5 + γ 0 / matrix. The normalizations, rojection oerators, roagators, dynamical invariants etc have been given in [5], for examle. We should assume the following relation in the field oerator (3: h= v h (b h ( = h= u h ( a h (. (5 We need Λ µλ ( = v µ (u λ (. In the sinorial basis by direct calculations, we find Λ µλ = im(σ n µλ, n = /, rovided that the normalization was chosen to the mass m. The indices h and σ, µ, λ are the corresonding olarization indices. However, in the helicity basis with the helicity oerator σ = ( cos θ sin θe iϕ sin θe +iϕ cos θ the -eigensinors can be defined as follows [6, 7]: φ = ( cos θ e iϕ/ sin θ e+iϕ/ (6 ( sin, φ = θ e iϕ/ cos θ, (7 e+iϕ/ for ±/ eigenvalues, resectively. We can start from the Klein-Gordon equation, generalized for describing the sin-/ articles (i. e., two degrees of freedom; c = h = : ( 0 + σ ( 0 σ φ = m φ. (8 3

4 If the φ sinors are defined by the equation (7 then we can construct the corresonding u and v 4-sinors: u ( = N + u ( = N + v ( = N v ( = N ( φ E m ( φ E + m ( φ E m ( φ E+ m = = = E+ φ m m φ, E + m φ E + E+ φ m E+ φ m m φ, E + m φ E + E+ φ m = where the normalization to the unit (± was now used:, (9, (0 ū h (u h ( = δ hh, v h (v h ( = δ hh, ( ū h (v h ( = 0 = v h (u h (. ( The commutation relations may be assumed to be the standard ones [4, 3, 8, 9] (comare with [0] [ ah (, a h (k] + = E δ (3 ( kδ hh, [a h (, a h (k] + = 0 = [ a h (, a h (k] + (3 [ ah (, b h (k] + = 0 = [ b h (, a h (k] +, (4 [ bh (, b h (k] + = E δ (3 ( kδ hh, [b h (, b h (k] + = 0 = [ b h (, b h (k] + (5 Other details of the helicity basis are given in Refs. [3,, ]. However, in this helicity case we construct Λ µλ ( = v µ (u λ ( = iσ y µλ. (6 It is well known that article=antiarticle in the Majorana theory [4]. So, in the language of the quantum field theory we should have b µ (E, = e iϕ a µ (E,. (7 4

5 Usually, different authors use ϕ = 0, ±π/ deending on the metrics and on the forms of the 4-sinors and commutation relations, etc.. The alication of the Majorana anzatz leads to the contradiction in the sinorial basis. Namely, it leads to existence of the referred axis in every inertial system (only y survives, thus breaking the rotational symmetry of the secial relativity. Next, we can use another Majorana anzatz Ψ = ±e iα Ψ c with usual definitions ( 0 iθ C = e iϑc K, Θ = iθ 0 ( 0. (8 0 Thus, on using Cu ( = iv (, Cu ( = +iv ( we come to other relations between creation/annihilation oerators a ( = ±ie iα b (, (9 a ( = ie iα b (, (0 which may be used instead of (7. In the case of α = π/ we have similar relations as in (6, but for ositive-energy oerators. Due to the ossible signs ± the number of the corresonding states is the same as in the Dirac case that ermits us to have the comlete system of the Fock states over the (/, 0 (0, / reresentation sace in the mathematical sense. Please note that the hase factors may have hysical significance in quantum field theories as oosed to the textbook nonrelativistic quantum mechanics, as was discussed recently by several authors. However, in this case we deal with the self/anti-self charge conjugate quantum field oerator instead of the self/anti-self charge conjugate quantum states. Please remember that it is the latter that answer for the neutral articles. The quantum field oerator contains oerators for more than one state, which may be either electrically neutral or charged. 3 Helicity Basis in the (, 0 + (0, Reresentation. The solutions of the Weinberg-like equation [γ µν µ ν (i / t E m ]Ψ(x = 0. ( 5

6 are found in Refs. [5, 6, 7, 8]. Here they are: ( D u σ ( = S ( (Λ R ξ σ (0 D D S, v (Λ L ξ σ (0 σ ( = S (Λ R Θ [/] ξσ(0 D S (Λ L Θ [/] ξσ(0 = Γ 5 u σ (, ( ( Γ =, ( in the sinorial reresentation. The D S is the matrix of the (S, 0 reresentation of the sinor grou SL(, c. In the (, 0 (0, reresentation the rocedure of derivation of the creation oerators (in the similar way as in the revious Section leads to somewhat different situation: Hence, σ=0,± v σ (b σ( = σ=0,± u σ ( a σ (. (4 b σ( 0. (5 However, if we return to the original Weinberg equations [γ µν µ ν ±m ]Ψ, (x = 0 with the field oerators: we obtain Ψ (x = Ψ (x = (π 3 µ (π 3 µ d 3 E [u µ (a µ (e i x + u µ (b µ(e +i x ], (6 d 3 E [v µ (c µ (e i x + v µ (d µ(e +i x ], (7 b µ( = [ (S n ] µλ a λ (, (8 d µ( = [ (S n ] µλ c λ (. (9 The alication of u µ ( u λ ( = δ µλ and u µ ( u λ ( = [ (S n ] µλ rove that the equations are self-consistent (similarly to the consideration of the (/, 0 (0, / reresentation. This situation signifies that in order to construct the Sankaranarayanan-Good field oerator (which was used by Ahluwalia, Johnson and Goldman [7] we need additional ostulates. One can try to construct the left- and the right-hand side of the field oerator 6

7 searately each other. In this case the commutation relations may also be more comlicated. Is it ossible to aly the Majorana-like anzatz to the (, 0+(0, fields? It aears that in this basis we also come to the same contradictions as before. We have two equations and a µ ( = +e iϕ [ (S n ] µλ a λ (, (30 a µ( = +e +iϕ [ (S n ] µλ a λ (. (3 In the basis where S z is diagonal the matrix S y is imaginary [6]. So, (S n = S x n x S y n y + S z n z, and (S n (S n in the case of S =. So, we conclude that there is the same roblem in this oint, in the alication of the Majorana-like anzatz, as in the case of sin-/. Similarly, one can roceed with (9. What we would have in the basis where all S i jk = iɛ ijk are ure imaginary? Finally, I just want to mention that the attemts of constructing the self/anti-self charge conjugate states failed in Ref. [9]. Instead, the Γ 5 S[] c self/anti-self conjugate states have been constructed therein. Now we turn to the helicity basis. The helicity oerator in the (/, / reresentation is frequently resented: (S I = 0 0 i 3 i 0 i 3 0 i, (S ɛ µ ± = ±ɛ µ (S ±, ɛ µ 0,0 t = 0. 0 i i 0 (3 However, we are aware about some roblems with the chosen basis. The helicity oerator is (in the case of S 3 diagonal: (S II = 0 z l 0 0 r 0 l. ( r z The unitary transformation [6,.55] i U = , U (SI U = (SII 0 + i 0 (34 7

8 can be erfomed to transfer oerators and olarization vectors from one basis to another. The first-basis eigenvectors are: ɛ µ + = e iα ɛ µ 0 = m E E E i ( +( 3 i ( +( ( + (,, ɛ µ = e iβ 0 3 +i ( +( 3 i ( +( ( + ( (35 E ɛµ 0 t = m. (36 3 The eigenvectors ɛ µ ± are not the eigenvectors of the arity oerator (γ 00 R of this reresentation. However, the ɛ µ,0, ɛ µ 0,0 t are. Surrisingly, the latter have no well-defined massless limit. In order to get the well-known massless limit one should use the basis of the light-front form rerersentation, cf. [0]. We also note that the olarization vectors have relations to the solutions of the (, 0 (0, reresentation through the Proca equations or the Duffin- Kemmer-Petiau equations. The corresonding helicity oerator of the (, 0 (0, reresentation is ĥ = The eigen 3-vectors are [6, 7] φ = N e iϑ + ( (S (S 3 3 ( + cos θe iϕ sin θ, φ = N e iϑ ( cos θe iϕ sin θ (37 ( cos θe+iϕ ( + cos θe+iϕ (38 sin θ e iϕ φ = N e iϑ 0 cos θ. (39 sin θ e+iϕ Finally, some notes concerning with the tachyonic solutions of the Weinberg equations in the (, 0 (0, reresentation sace. While some authors, 8

9 e.g. Ref. [], argued recently that the tachyonic energy-momentum relation E = ± c m 0c 4 may lead to some interretational roblems, we still consider it in this aer. The Weinberg equations [γ µν µ ν ± m ]Ψ, (x = 0 give us both bradyonic and tachyonic solutions, E = ± c ± m 0c 4. We resent them now in the helicity basis which may hel us to overcome the difficulties in the construction of the Majoran(-like field oerators, as shown above. If φ R (0 = φ L (0 the 6-objects can be normalized to the unit. The solutions of [γ µν µ ν m ]Ψ(x = 0 are u ( = ( m χ E E χ m, u ( = ( m χ E + E + χ m, (40 u ( = ( χ. (4 In the case of tachyonic solutions (E < we shall be no able to normalize to. However, it is ossible to normalize to -. In this case we have in the helicity basis: U ( = ( E+ χ m m E + χ χ, U ( = ( U ( = ( χ χ m χ E + E+ χ m, (4. (43 Nevertheless, self/anti-self charge-conjugated 6-objects have not been constructed till now. 4 Conclusions. We conclude that the calculations in the helicity basis may be useful to give mathematically and hysically reasonable results when dealing with the Majorana articles. Acknowledgements. I acknowledge discussions with colleagues at recent conferences. I am grateful to the Zacatecas University for rofessorshi. References [] V. V. Dvoeglazov, Hadronic J. Sul. 8, 39 (003. 9

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