aa r X i v : . [ phy s i c s . g e n - ph ] J u l A new class of mass dimension one fermions
Dharam Vir Ahluwalia a, ∗ a Mountain Physics Camp, Center for the Studies of the Glass Bead GameBir, Himachal Pradesh, 176077 India
Abstract
These are notes on the square root of 4 × p µ p µ = m [1] 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 the new spin one-half fermions with mass dimension one provides astrong reason that other roots of I may lead to new spin one half matter fields, and thesemay serve the dark matter sector or at the least provide us with a complete set of particlecontent consistent with basic principles of quantum mechanics and symmetries of specialrelativity [4–19].With this background and motivation we recall the well known linearly independent ∗ Corresponding author
Email address: [email protected] (Dharam Vir Ahluwalia)
Preprint submitted to Proceedings of the Royal Society A 04 July 2020 quare roots of identity [20, p. 71] I (1) iγ iγ iγ γ (2) iγ γ iγ γ iγ γ γ γ γ γ γ γ (3) iγ γ γ iγ γ γ iγ γ γ γ γ γ (4) iγ γ γ γ (5)We denote these by Γ ℓ , ℓ = 1 , · · ·
16, with Γ being the first entry in the above array andΓ being the last – ℓ assignment is in consecutive order. We shall adopt the Weyl basis inthe (1 / , ⊕ (0 , /
2) representations space.To illustrate the method we consider Γ . Its eigenspinors, up to constant multiplicativefactors, are λ = − i , λ = i , λ = − i , λ = i . (6)The first and the third eigenspinors correspond to eigenvalue +1 of Γ , and the other twoto eigenvalue − . We define these as the ‘rest spinors’ λ i (0).By acting the boost operator κ = r E + m m " I + σ · p E + m I − σ · p E + m (7)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 λ ( p ) = 0 (8) m − γ µ p µ λ ( p ) − τ λ ( p ) = 0 , m − γ µ p µ λ ( p ) − τ λ ( p ) = 0 (9)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 instead 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, the 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 − I (cid:19) (10)2here ǫ µνλσ is the completely antisymmetric 4th rank tensor with ǫ = +1 (the dimen-sionality of identity matrix I and null matrix shall be apparent from the context)]. Thecharge conjugation C , parity P , and time reversal T , operators can then be written as C = (cid:18) i Θ − i Θ (cid:19) K, P = m − γ µ p µ , T = iγ C (11)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 ) , (12) P λ ( p ) = λ ( p ) , P λ ( p ) = λ ( p ) , P λ ( p ) = λ ( p ) , P λ ( p ) = λ ( p ) , (13) T λ ( p ) = − iλ ( p ) , T λ ( p ) = iλ ( p ) , T λ ( p ) = − iλ ( p ) , T λ ( p ) = iλ ( p ) (14)with the consequence that ( CPT ) = I , with C = I , P = I , T = − I . The chargeconjugation and parity operators anticommute: {C , P} = 0 . As in the case for Elko [4, 21], here too we find that under the Dirac dual each of the λ i ( p ), i = 1 , , ,
4, has null norm. As such we define a new dual: ¬ λ ( p ) = (cid:2) + P λ ( p ) (cid:3) † γ = λ ( p ) , ¬ λ ( p ) = (cid:2) + P λ ( p ) (cid:3) † γ = λ ( p ) , (15) ¬ λ ( p ) = (cid:2) − P λ ( p ) (cid:3) † γ = − λ ( p ) , ¬ λ ( p ) = (cid:2) − P λ ( p ) (cid:3) † γ = − λ ( p ) . (16)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 , ¬ λ i ( p ) λ i ( p ) = − m, i = 3 , X i =1 , λ i ( p ) ¬ λ i ( p ) = 2 m , X i =3 , λ i ( p ) ¬ λ i ( p ) = − m (19)leading to the completeness relation12 m X i =1 , λ i ( p ) ¬ λ i ( p ) − X i =3 , λ i ( p ) ¬ λ i ( p ) = I . (20) The freedom in the definition of spinorial duals was first pointed out in an unpublished e-print of theauthor [22], and after several intervening publications it takes its final form for Elko in [5]. The subjecthas now developed into a research sub-field of its own. We refer the reader to [23] for a sense of excitementand relevant references.
3e thus introduce a new spin one half quantum 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 ) e − ip · x + X i =3 , b † i ( p ) λ i ( p ) e ip · x (cid:21) (21)with ¬ b ( x ) def = Z d p (2 π ) p mE ( p ) (cid:20) X i =1 , a † i ( p ) ¬ λ i ( p ) e ip · x + X i =3 , b i ( p ) ¬ λ i ( p ) e − ip · x (cid:21) (22)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 (23)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 (24)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) (25)where— the plus sign holds for bosons and the minus sign for 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 ) (26) 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 ) . (27)4he two Heaviside step functions of equation (25) can now be replaced by their integralrepresentations θ ( t ′ − t ) = lim ǫ → + Z d ω πi e iω ( t ′ − t ) ω − iǫ (28) θ ( t − t ′ ) = lim ǫ → + Z d ω πi e iω ( t − t ′ ) ω − iǫ (29)where ǫ, ω ∈ R . Using these results, and • substituting ω → p = − ω + E ( p ) in the first term and ω → p = ω − E ( p ) in thesecond term • and using the results (20) for the spin sumswe are forced – by internal consistency of the resulting formalism – to pick the minus signin (25), 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 (30)This is equivalent to the choice (23) over (24). Following [4], the normalisation ξ is seento 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 (32)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 (33)so that (cid:16) ∂ µ ′ ∂ µ ′ I + m I (cid:17) S FD ( x ′ − x ) = − δ ( x ′ − x ) (34)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 (35)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 ) (36)5his determines the momentum conjugate to b ( x ) π ( x ) = ∂ L ( x ) ∂ ˙ b ( x ) = ∂∂t ¬ b ( x ) . (37)Using the spin sums given in equation (19) we determine the locality structure of the newfermionic field to be (cid:8) b ( t, x ) , π ( t, x ′ ) (cid:9) = iδ (cid:0) x − x ′ (cid:1) I , (38) (cid:8) b ( t, x ) , b ( t, x ′ ) (cid:9) = 0 , (cid:8) π ( t, x ) , π ( t, x ′ ) (cid:9) = 0 . (39)To examine if the energy associated with the introduced b ( x )- ¬ b ( x ) system has the usualzero point contribution and is bounded from below, we carry out a calculation similar tothe one presented in [24, Section 7] and find the field energy to be H = Z d p (2 π ) m E ( p ) " X i =1 , a † i ( p ) a i ( p ) ¬ λ i ( p ) λ i ( p ) + X i =3 , b i ( p ) b † i ( p ) ¬ λ i ( p ) λ i ( p ) (40)Use of the orthonormality relations (17) and (18) reduce the above expression to H = Z d p (2 π ) E ( p ) " X i =1 , a † i ( p ) a i ( p ) − X i =3 , b i ( p ) b † i ( p ) (41)Consistent with the obtained fermionic locality anticommutator (38), the next simplifica-tion occurs by exploiting { b i ( p ) , b i ′ ( p ′ ) } = (2 π ) δ ( p − p ′ ) δ ii ′ (42)with the result that H = − δ ( ) Z d p E ( p ) | {z } H + X i =1 , Z d p (2 π ) E ( p ) a † i ( p ) a i ( p ) + X i =3 , Z d p (2 π ) E ( p ) b † i ( p ) b i ( p )To obtain a representation for δ ( ) that appears in the above expression for the fieldenergy, we note that since δ ( p ) may be expanded as δ ( p ) = 1(2 π ) Z d x exp( i p · x ) (43) δ ( ) may be replaced by [1 / (2 π ) ] R d x , giving the following contribution for the zeropoint energy H = − × π ) Z d x Z d p E ( p ) (44)6ince in natural units ~ is set to unity, π ) d x d p acquires the interpretation of a unit-sizephase cell, with − E ( p ) as its energy content. The factor of 4 in the expression for H corresponds to the four particle and antiparticle degrees carried by the b ( x )- ¬ b ( x ) system.The remaining two terms in the expression for H establish that for a given momentum p each of the four particle-antiparticle degrees of freedom contributes equally.This completes our construction of an entirely new class of spin one half fermions.Their physical implications are essentially unknown. Because of the mass dimensionalitymismatch with the standard model fermions – 3 / Note.
Because the results presented here have evolved out of a manuscript arXived ase-print [25] it is important to make a remark. The point of departure starts with equa-tion (16), that is the definitions of the duals of λ ( p ) and λ ( p ). This change percolatesthrough the rest of the calculations, finally replacing (cid:2) b ( t, x ) , p ( t, x ′ ) (cid:3) = iδ (cid:0) x − x ′ (cid:1) I ℓ , (cid:2) b ( t, x ) , b ( t, x ′ ) (cid:3) = 0 , (cid:2) p ( t, x ) , p ( t, x ′ ) (cid:3) = 0 . where I ℓ def = − − of the e-print by equations (38) and (39). This noted, if one proceeds with commutatorcounterparts of (42), and keeps the duals of λ ( p ) and λ ( p ) as before, the resulting fieldenergy is found to be same as above but with H replaced by H = +4 × π ) Z d x Z d p E ( p ) . Funding.
The research presented here is entirely supported by the personal funds of theauthor.
Acknowledgements.
I am grateful to the two anonymous referees who reviewed andcommented constructively. I thank Julio Marny Hoff da Silva and Cheng-Yang Lee forcorrespondence related to the ideas presented here, and Sweta Sarmah for discussions.
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