aa r X i v : . [ phy s i c s . g e n - ph ] A ug Theory of spin one half bosons
Dharam Vir Ahluwalia a, ∗ a International Centre for Cosmology, P. D. Patel Institute of Applied Sciences, Charotar University ofScience & Technology, Changa, Gujarat 388421, India
Abstract
These are notes on the square root of 4 × p µ p µ = m The square root of the left hand side was found to be γ µ p µ , where the γ µ arethe celebrated 4 × γ µ p µ ± m I ) ψ ( p ) = 0, the Dirac equation in momentum space. Its solutions, after attendingto certain locality phases, later became expansion coefficients of all fermionic matter fieldsof the standard model [2].Modulo the Majorana observation of 1937 [3], there is a general consensus that theDirac field presents a unique spin one half field that is consistent with Lorentz symmetriesand locality. The uniqueness, however, hinges on the implicit assumption that the squareroot of a 4 × I multiplying the m on the right hand side, is I itself. Therecent emergence of new spin one-half fermions with mass dimension one provides a strongreason that other roots of I may lead to new spin one half matter fields, and these mayserve the dark matter sector or at the least us provide us a complete set of particle contentconsistent with basic principles of quantum mechanics and Lorentz symmetries [4].With this background and motivation we here present non-trivial square roots A of I : AA def = I . where A is a 4 × ∗ Corresponding author
Email address: [email protected] (Dharam Vir Ahluwalia)
Preprint submitted to Yet to be Submitted August 27, 2019 s antisymmetric and symmetric – the most general roots being deferred to our readers .A detailed calculation shows that these depend only on A
13 def = γ + iδ and A
14 def = α + iβ with α, β, γ, δ ∈ R . Our results on the roots are collected together in Appendix A. Unlikethe Dirac root I none of the twenty eight roots we find commute with γ µ p µ . This non-commutativity thus requires us to examine the behaviour of the eigenvectors of A underthe action of γ µ p µ . For Lorentz symmetries not to be violated if λ i , i = 1 , , ,
4, are thementioned eigenspinors Aλ i ( p ) = aλ i ( p ) , a ∈ R (1)then the action of m − γ µ p µ on λ i ( p ), up to a factor of ± i or ±
1, must be one of theeigenspinors, λ j = i ( p ).To illustrate the method I consider the first of the two A ’s in equation (A.1). Its eigen-spinors, up to constant factors, are λ = − i , λ = i , λ = − i , λ = i (2)The first and the third eigenspinors correspond to eigenvalue +1 of A , and the other twoto eigenvalue − A . We define these as the ‘rest spinors’ λ i (0). By acting the boostoperator κ = r E + m m " I + σ · p E + m OO I − σ · p E + m (3)on these spinors we obtain the four eigenspinors for an arbitrary momentum λ i ( p ) = κλ i (0).We implement our programme by solving the following four equations for τ ij ∈ R : m − γ µ p µ λ ( p ) − τ λ ( p ) = 0 , m − γ µ p µ λ ( p ) − τ τ λ ( p ) = 0 (4) m − γ µ p µ λ ( p ) − τ λ ( p ) = 0 , m − γ µ p µ λ ( p ) − τ λ ( p ) = 0 (5)and find that a single τ , equal to unity, solves all the four equations and assures that while λ i ( p ) do not satisfy the Dirac equation they certainly satisfy the spinorial Klein-Gordonequation. We thus pass the first test for the viability of the theory to be Lorentz covariant.To study the CPT properties of λ ( p ) we introduce Θ (Wigner time reversal operator)and γ Θ = (cid:18) −
11 0 (cid:19) , γ = i ǫ µνλσ γ µ γ ν γ λ γ σ = (cid:18) I OO − I (cid:19) (6) We work in Weyl representation. ǫ µνλσ is the completely antisymmetric 4th rank tensor with ǫ = +1. The chargeconjugation C , parity P , and time reversal T , operators can then be written as C = (cid:18) O i Θ − i Θ O (cid:19) K, P = m − γ µ p µ , T = iγ C (7)where K complex conjugates to the right. We then readily obtain C λ ( p ) = − λ ( p ) , C λ ( p ) = λ ( p ) , C λ ( p ) = λ ( p ) , C λ ( p ) = − λ ( p ) , (8) P λ ( p ) = λ ( p ) , P λ ( p ) = λ ( p ) , P λ ( p ) = λ ( p ) , P λ ( p ) = λ ( p ) , (9) T λ ( p ) = − iλ ( p ) , T λ ( p ) = iλ ( p ) , T λ ( p ) = − iλ ( p ) , T λ ( p ) = iλ ( p ) (10)with the consequence that ( CPT ) = I , with C = I , P = I , T = − I . And, {C , P} = 0 . As in the case for Elko [4], here too we find that under the Dirac dual the λ ( p ) havenull norm. As such we define a new dual: ¬ λ ( p ) = λ ( p ) , ¬ λ ( p ) = λ ( p ) , ¬ λ ( p ) = λ ( p ) , ¬ λ ( p ) = λ ( p ) (11)After re-norming the rest eigenspinors by a multiplicative factor of √ m , the new dual givesthe following Lorentz invariant orthonormality relations ¬ λ i ( p ) λ i ( p ) = +2 m, i = 1 , , (12) ¬ λ i ( p ) λ i ( p ) = +2 m, i = 3 , X i =1 , λ i ( p ) ¬ λ i ( p ) = 2 m , X i =3 , λ i ( p ) ¬ λ i ( p ) = 2 m (14)leading to the completeness relation X i =1 , λ i ( p ) ¬ λ i ( p ) + X i =3 , λ i ( p ) ¬ λ i ( p ) = 2 m I (15)The appearance of the plus, rather than minus, sign between the two terms above wouldeventually justify the title of this communication. We thus introduce a new spin one halfquantum field with λ i ( p ) as its expansion co-efficients: b ( x ) def = Z d p (2 π ) p mE ( p ) (cid:20) X i =1 , a i ( p ) λ i ( p ) (cid:3) e − ip · x + X i =3 , b † i ( p ) λ i ( p ) (cid:3) e ip · x (cid:21) (16)3ith ¬ b ( x ) def = Z d p (2 π ) p mE ( p ) (cid:20) X i =1 , a † i ( p ) ¬ λ i ( p ) (cid:3) e ip · x + X i =3 , b i ( p ) ¬ λ i ( p ) (cid:3) e − ip · x (cid:21) (17)as its adjoint. At this stage we do not fix the statistics to be fermionic n a i ( p ) , a † j ( p ) o = (2 π ) δ ( p − p ′ ) δ ij , (cid:8) a i ( p ) , a j ( p ′ ) (cid:9) = 0 = n a † i ( p ) , a † j ( p ′ ) o (18)or bosonic h a i ( p ) , a † j ( p ) i = (2 π ) δ ( p − p ′ ) δ ij , (cid:2) a i ( p ) , a j ( p ′ ) (cid:3) = 0 = h a † i ( p ) , a † j ( p ′ ) i (19)and assume similar anti-commutation, or commutation, relations for b i ( p ) and b † i ( p ).To determine the statistics for the b ( x ) and ¬ b ( x ) system we consider two events, x and x ′ , and note that the amplitude to propagate from x to x ′ is then A x → x ′ = ξ (cid:16) h | b ( x ′ ) ¬ b ( x ) | i θ ( t ′ − t ) ± h | ¬ b ( x ) b ( x ′ ) | i θ ( t − t ′ ) | {z } h | T ( b ( x ′ ) ¬ b ( x ) | i (cid:17) (20)where— the plus sign holds for the bosons and the minus sign for the fermions,— ξ ∈ C is to be determined from the normalisation condition that A x → x ′ integratedover all possible separations x − x ′ be unity (or, more precisely e iγ , with γ ∈ R ).— and T is the time ordering operator.The two vacuum expectation values that appear in A x → x ′ evaluate to the following expres-sions h | b ( x ′ ) ¬ b ( x ) | i = Z d p (2 π ) (cid:18) mE ( p ) (cid:19) e − ip · ( x ′ − x ) X i =1 , λ i ( p ) ¬ λ i ( p ) (21) h | ¬ b ( x ) b ( x ′ ) | i = Z d p (2 π ) (cid:18) mE ( p ) (cid:19) e ip · ( x ′ − x ) X i =3 , λ i ( p ) ¬ λ i ( p ) . (22)The two Heaviside step functions of equation (20) can now be replaced by their integralrepresentations θ ( t ′ − t ) = lim ǫ → + Z d ω πi e iω ( t ′ − t ) ω − iǫ (23) θ ( t − t ′ ) = lim ǫ → + Z d ω πi e iω ( t − t ′ ) ω − iǫ (24)where ǫ, ω ∈ R . Using these results, and 4 substituting ω → p = − ω + E ( p ) in the first term and ω → p = ω − E ( p ) in thesecond term • and using the results (15) for the spin sumswe are forced – by internal consistency of the resulting formalism – to pick the plus signin (20), giving A x → x ′ = i ξ Z d p (2 π ) e − ip µ ( x ′ µ − x µ ) I p µ p µ − m + iǫ (25)This is equivalent to the choice (19) over (18). The normalisation ξ is seen to be [4] ξ = im A x → x ′ = − m Z d p (2 π ) e − ip µ ( x ′ µ − x µ ) I p µ p µ − m + iǫ (27)We define the Feynman-Dyson propagator S FD ( x ′ − x ) def = − m A x → x ′ = Z d p (2 π ) e − ip µ ( x ′ µ − x µ ) I p µ p µ − m + iǫ (28)so that (cid:16) ∂ µ ′ ∂ µ ′ I + m I (cid:17) S FD ( x ′ − x ) = − δ ( x ′ − x ) (29)In terms of the new field b ( x ) and its adjoint ¬ b ( x ) it takes the form S FD ( x ′ − x ) = − i D (cid:12)(cid:12)(cid:12) T ( b ( x ′ ) ¬ b ( x ) (cid:12)(cid:12)(cid:12) E (30)and establishes the mass dimension of the field to be one, leading to the following free fieldLagrangian density L ( x ) = ∂ µ ¬ b ∂ µ b ( x ) − m ¬ b ( x ) b ( x ) (31)This leads to the momentum conjugate to b ( x ) p ( x ) = ∂ L ( x ) ∂ ˙ b ( x ) = ∂∂t ¬ b ( x ) . (32)Using the spin sums given in equation (14) we determine the locality structure of the newbosonic field to be (cid:2) b ( t, x ) , p ( t, x ′ ) (cid:3) = iδ (cid:0) x − x ′ (cid:1) I ℓ , (33) (cid:2) b ( t, x ) , b ( t, x ′ ) (cid:3) = 0 , (cid:2) p ( t, x ) , p ( t, x ′ ) (cid:3) = 0 . (34)5here I ℓ def = − − (35)Had we taken the locality phases to be − λ and λ , the entire formalismwould have run precisely as above with the difference that the locality structure of the newbosonic field would have changed to (cid:2) b ( t, x ) , p ( t, x ′ ) (cid:3) = iδ (cid:0) x − x ′ (cid:1) I r , (36) (cid:2) b ( t, x ) , b ( t, x ′ ) (cid:3) = 0 , (cid:2) p ( t, x ) , p ( t, x ′ ) (cid:3) = 0 . (37)where I r def = − − (38)We thus conclude that in order to preserve locality parity must be maximally violated withthe left and right projections of b ( x ) serving as two independent local fields. This makesus suspect that neutrinos may be bosonic rather than fermionic as was first speculated byDolgov and Smirnov [5]. Acknowledgement —The author wishes to thank Suresh Chand for discussions. The constant factors mentioned just above equation (2). ppendix A. Square roots of 4 × Antisymmetric solutions, up to a sign A = − i i − i i , A = − i i i − i (A.1)With the definitions ǫ def = α + iβ (A.2) ε def = i p ( α + iβ ) + 1 = i p ǫ + 1 (A.3) η def = γ + iδ (A.4) ζ def = i p ( γ + iδ ) + 1 = i p η + 1 (A.5) ξ def = i p ( γ + iδ ) + ( α + iβ ) + 1 = i p η + ǫ + 1 (A.6)we have the following twelve additional A ’s (again, up to a sign) A = ε ǫ − ε − ǫ ǫ − ε − ǫ ε , A = − ε ǫε − ǫ ǫ ε − ǫ − ε (A.7) A = ε ǫ − ε ǫ − ǫ ε − ǫ − ε , A = − ε ǫε ǫ − ǫ − ε − ǫ ε (A.8) A = ζ η − ζ − η − η ζ η − ζ , A = − ζ η ζ − η − η − ζ η ζ (A.9) A = ζ η − ζ η − η − ζ − η ζ , A = − ζ η ζ η − η ζ − η − ζ (A.10)7 = ξ η ǫ − ξ − ǫ η − η ǫ − ξ − ǫ − η ξ , A = − ξ η ǫξ − ǫ η − η ǫ ξ − ǫ − η − ξ , (A.11) A = ξ η ǫ − ξ ǫ − η − η − ǫ ξ − ǫ η − ξ , A = − ξ η ǫξ ǫ − η − η − ǫ − ξ − ǫ η ξ (A.12)Symmetric solutions (restricted to diag { , , , } ), up to a sign: A = − − −