Spin one matter elds. November 2015

Transcription

1 Spin one matter elds M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, S. Gomez-Ávila Universidad de Guanajuato Mexican Workshop on Particles and Fields November 2015 M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 1) de GuanajuatoNovember Mexican Workshop 2015 on 1 / Partic 26

2 Spin one matter elds M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, S. Gomez-Ávila Universidad de Guanajuato Mexican Workshop on Particles and Fields November 2015 M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 2) de GuanajuatoNovember Mexican Workshop 2015 on 2 / Partic 26

3 Outline of the Talk 1 Motivations. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 3) de GuanajuatoNovember Mexican Workshop 2015 on 3 / Partic 26

4 Outline of the Talk 1 Motivations 2 The HLG and Poincaré.. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 3) de GuanajuatoNovember Mexican Workshop 2015 on 3 / Partic 26

5 Outline of the Talk 1 Motivations 2 The HLG and Poincaré. 3 Spin 1 algebra and equations of motion in momentum space.. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 3) de GuanajuatoNovember Mexican Workshop 2015 on 3 / Partic 26

6 Outline of the Talk 1 Motivations 2 The HLG and Poincaré. 3 Spin 1 algebra and equations of motion in momentum space. 4 Dynamics and constrictions. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 3) de GuanajuatoNovember Mexican Workshop 2015 on 3 / Partic 26

7 Outline of the Talk 1 Motivations 2 The HLG and Poincaré. 3 Spin 1 algebra and equations of motion in momentum space. 4 Dynamics and constrictions 5 Quantum Field Theory. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 3) de GuanajuatoNovember Mexican Workshop 2015 on 3 / Partic 26

8 Outline of the Talk 1 Motivations 2 The HLG and Poincaré. 3 Spin 1 algebra and equations of motion in momentum space. 4 Dynamics and constrictions 5 Quantum Field Theory 6 Conclusions and remarks M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 3) de GuanajuatoNovember Mexican Workshop 2015 on 3 / Partic 26

9 We may need to look in other direction to extend the SM M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 4) de GuanajuatoNovember Mexican Workshop 2015 on 4 / Partic 26

10 Fields transform under th HLG M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 5) de GuanajuatoNovember Mexican Workshop 2015 on 5 / Partic 26

11 Fields transform under th HLG They can be used in eective theories of compound systems (R χ PT, hadron physics). They can give alternative routes to study dark matter. Possible extensions to the standard model. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 6) de GuanajuatoNovember Mexican Workshop 2015 on 6 / Partic 26

12 Fields transform under th HLG The Poincaré algebra has two algebraic invariants C 2 = P µ P µ C 4 = W µ W µ with W µ = 1 2 ε µστρm στ P ρ One particle state satisfy C 2 Ψ = m 2 Ψ C 4 Ψ = m 2 j(j + 1) Ψ where we call m the mass and j the spin of Ψ. The quantum elds, the basic elements of a QFT allow us to calculate expectation values, are built from operators that create or destroy this states ˆ [ ] Ψ l = dγ e ipx ω l (Γ) a (Γ) + e ipx ωl c (Γ) a (Γ) the eld coecients ω, transform in the representations of the Lorentz group. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 7) de GuanajuatoNovember Mexican Workshop 2015 on 7 / Partic 26

13 HLG and parity The HLG is an homomorphism of SU (2) SU (2). Thus the representations can be labeled by two angular momenta (j A, j B ). But, under parity (j A, j B ) (j B, j A ). To have a state with well dened parity we must extend our space to (j A, j B ) (j B, j A ). Then to describe high spin matter elds we choose j A = j and j B = 0. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 8) de GuanajuatoNovember Mexican Workshop 2015 on 8 / Partic 26

14 Covariant basis It was proven by S. Gomez and M. Napsuciale 1 that the parity based covariant basis for a general(j, 0) (0, j)contains: Two Lorentz scalars. Six operators transforming in (1, 0) (0, 1) forming a second rank antysimmetric tensor. A pair of symmetric traceless matrices S µ 1µ 2...µ j A series of matrix tensor operators, wich transform in the representation (2, 0) (0, 2), (3, 0) (0, 3),... (2j, 0) (0, 2j) /PhysRevD M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 9) de GuanajuatoNovember Mexican Workshop 2015 on 9 / Partic 26

15 The ( 1 2, 0) ( 0, 1 2) equation of motion in momentum space: Dirac Equation. As an example let us take j = 1 2. Now take the projection over parity eigenstates Πu (0) = ±u (o) Now we can apply a boost to this equation to obtain ( B (p) ΠB 1 (p) ± 1 ) u (p) = 0 it turns out to be that then we recover the Dirac equation B (p) ΠB 1 (p) = γµ p µ m (γ µ p µ ± m) u (p) = 0 in principle we can apply the same procedure for dierent spins. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 10) de Guanajuato November Mexican 2015 Workshop10 on/ Partic 26

16 The (1, 0) (0, 1)equation of motion in momentum space. We can get the equation of motion by boosting the rest-frame parity-projection, but now the elds transform in the representation (1, 0) (0, 1)of the LG. This will give us ( S µν ) p µ p ν ± I ψ (p) = 0 Λ ± ψ (p) = ψ (p), m 2 where S µν is a traceless tensor of rank 2 and Λ ± ± 1 ( S µν p µ p ν 2 is the projector. m 2 ) ± I, M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 11) de Guanajuato November Mexican 2015 Workshop11 on/ Partic 26

17 We have to be careful when the system is out of shell.. To get the projector out of the mass shell we replace m 2 by p 2 ( 1 S µν p µ p ν 2 p 2 ) I ψ (p) = ψ (p), and to have a local theory we project over the Poincaré orbit p 2 = m 2 so we obtain: 1 2 (S µν p µ p ν η µν p µ p ν ) ψ (p) = m 2 ψ (p), and if we dene a new operator Σ µν 1 2 (S µν η µν ) we obtain: ( Σ µν p µ p ν ± m 2) u (p) = 0. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 12) de Guanajuato November Mexican 2015 Workshop12 on/ Partic 26

18 The S tensor have some interesting properties. S µν fullls some Jordan algebra, which is analogous to the Cliord algebra of the γ µ in Dirac theory: { S µν, S αβ} = 4 (η µα η νβ + η να η µβ 12 ) 3 ηµν η αβ 1 6 ( C µανβ + C µβνα), the tensor C µανβ satises C µανβ = C αµνβ = C αµβν, C µανβ = C νβµα and the Bianchi identity. The commutator is, on the other hand: [ S µν, S αβ] = i ( η µα M νβ + η να M µβ + η νβ M µα + η µβ M να). It is clear from here that S 2 (p) S µν S αβ p µ p ν p α p β = p 4, analogous to γ µ γ ν p µ p ν = p 2 for Dirac. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 13) de Guanajuato November Mexican 2015 Workshop13 on/ Partic 26

19 We choose the parity basis for S. To study the dynamics of our equations we need to write the S µν in some specic basis, for simplicity and clarity we choose the parity basis: ) ( 0 J i S 00 = Π = S ij = ( I 0 0 I S 0i = J i I ( η ij + { J i, J j} 0 0 η ij { J i, J j} ), where J i = 1 2 ɛ ijkm jk are the conventional spin one matrices. ), M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 14) de Guanajuato November Mexican 2015 Workshop14 on/ Partic 26

20 The Lagrangian of the theory. As we have seen previously the momentum space equation of motion is ( Σ µν p µ p ν m 2) u (p) = 0, in conguration space this would read ( Σ µν µ ν + m 2) Ψ (x) = 0, from here It turns out that we can get this equation of motion from L = µ ΨΣ µν ν Ψ m 2 ΨΨ. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 15) de Guanajuato November Mexican 2015 Workshop15 on/ Partic 26

21 Using the explicit representation. To use the representation of the S it is convenient to write Ψ as Ψ = ( φ ξ where the canonical momentum are and π a = τ a = ), ς = ( π, τ ) δl δ ( 0 φ a ) = φ a 1 2 ( i ξ J i) a, δl ( δ ( 0 ξ a ) = 1 i ξ J i) 2, a form here it is clear that we have the restrictions: ρ a = τ a + 1 ( i ξ J i) ρ a = τ a + 1 ( J i i ξ ) 2 a 2 a. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 16) de Guanajuato November Mexican 2015 Workshop16 on/ Partic 26

22 The new Hamiltonian with constraints. Now, following Dirac, the time evolution of the system is given by H dened as ˆ H = d 3 xh + λ a ρ a + λ aρ a. The Hamilton equations that are modied with this change of Hamiltonian are: 0 τ a = δh δξ a 0 ξ a = δh δτ a = 1 2 i = λ a ( πj i ) a 3 4 ( i j ξ J i J j) + m 2 ξ a, M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 17) de Guanajuato November Mexican 2015 Workshop17 on/ Partic 26

23 Secondary constraints. In our particular case we dene the Possion brackets as ˆ [ δa (x) {A (x), B (y)} = d 3 δb (y) x δψ a δς a It is very easy to prove that δa (y) δς a ] δb (x), δψ a {φ a (x), π b (y)} = δ ab δ 3 (x y) {ξ a (x), τ b (y)} = δ ab δ 3 (x y) The constraints that we obtained before must be satised at any time this implies that { o ρ ( ) a = ρ ( ) a, H } = 0 This will produce secondary constraints in our theory χ ( ) a = i (πj i ) ( ) a 1 ( i j ξ J i J j) ( ) + m 2 ξ a ( ) = 0. 2 a there are not any other secondary constraints. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 18) de Guanajuato November Mexican 2015 Workshop18 on/ Partic 26

24 Poisson Brackets. Accordingly to Dirac we must calculate the matrix of Poisson brackets ab (x, y) = {f a (x), f b (y)} = m 2 δ 3 (x y) which has an inverse (this implies that the constraints are second class): ab (y, z) == 1 m 2 δ3 (y z) To go from classical to quantum mechanics we perform the transformation {A, B} D i [A, B] where ˆ {A, B} D = {A, B} d 3 zd 3 y {A, f a (z)} 1 ab (z, y) {f b (y), A}. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 19) de Guanajuato November Mexican 2015 Workshop19 on/ Partic 26

25 Commutation relations for the theory. The Possion brackets for the elds and their respective momentum are [ ] (J )2 {φ a (x), π b (y)} D = 1 2m 2 δ 3 (x y), {ξ a (x), τ b (y)} D = (J )2 ab 2m 2 δ 3 (x y) {ξ a (x), π b (y)} D = {φ a (x), τ b (y)} D = 0 and in a spinorial language If we calculate [ {Ψ a (x), ς b (y)} D = Σ 00 ab (J )2 2m 2 S 00 ] ab δ 3 (x y), then to go to the quantum theory we only must include a i. and equate this to the commutator. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 20) de Guanajuato November Mexican 2015 Workshop20 on/ Partic 26

26 Fourier Expansion The rst step of canonical quantization is to expand the elds as a Fourier series: Ψ(x) = α(p) [ c r (p)u r (p)e ipx + d r + (p)ur c (p)e ipx] p,r know we calculate all the physical quantities by imposing the usual commutation relations to the coecients [ ] [ ] c r (p), c s (p) = δ rs δ pp d r (p), d s (p) = δ rs δ pp now we can calculate the conjugated momenta which turns out to be ς d = ς d = L Ψ d,0 = Σ 0µ da ( µψ) a L = ( µ Ψ ) Ψ a Σ0µ ad d,0 M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 21) de Guanajuato November Mexican 2015 Workshop21 on/ Partic 26

27 Commutation relations Using the on shell projector we get the following result for the equal time commutation relations [ς d (x1), Ψ b (x2)] x 0 12 =0 = i p µ Λ (p) 2Vp ba Σ µ0 ad (e ip i(x1 i xi 2) e ip i(x1 i xi 2) ) p 0 now changing p p in the second term and using the algebra of the S tensor we get [ς d (x1), Ψ b (x2)] x 0 12 =0 = i ( e ip ( i(x1 i xi 2) S Σ 00 ij + g ij S 00) ) + V 4m 2 p i p j p making use again of the algebra we get nally [ς d (x1), Ψ b (x2)] x 0 12 =0 = i ( ) Σ 00 + (J p)2 S 00 e ip i(x1 i xi 2) 2m 2 V p ( ) [ς d (x1), Ψ b (x2)] x 0 12 =0 = i Σ 00 (J )2 S 00 2m 2 δ 3 (x1 x2) M. Napsuciale, this is just S. Rodriguez, the expected R.Ferro-Hernández, result.. Spine S. Gomez-Ávila one elds(slide (Universidad 22) de Guanajuato November Mexican 2015 Workshop22 on/ Partic 26

28 Energy-momentum and charge of the eld The energy momentum tensor and current are obtained as usual T ν µ = ν ΨΣ µα α Ψ + α ΨΣ αµ ν Ψ η ν µ ( α ΨΣ αν α Ψ m 2 ΨΨ ) J α = iq ( ( µ Ψ)S µα Ψ ΨS αν ( ν Ψ) ) By substituting the Fourier expansion in this expression we have proved that P µ = p,r [c + r (p)c r (p) + d + r (p)d r (p)]p µ Q = q p,r ( d + r (p)d r (p) c + r (p)c r (p) ) which is the expected result form a well behaved theory. It is important to remark that some factors are only reduced using the algebra of the S tensor. M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 23) de Guanajuato November Mexican 2015 Workshop23 on/ Partic 26

29 2-point Green Function The two point Green Function iγ F (x y) ab is the time ordered vacuum expectation value of the elds at dierent spacetime points. for our elds we have we have { iγ F (x y) ab = Γ F (x y) ab 0 T { φ a (x) φ b (y) } 0 p p 1 2V ω p Λ (p) ab e ip i(x1 i xi 2) x 0 > y 0 1 2V ω p Λ (p) ab e ip i(x1 i xi 2) y 0 > x 0 M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 24) de Guanajuato November Mexican 2015 Workshop24 on/ Partic 26

30 2-point Green Function After going to the complex plane we get that the propagator obtained from quantum eld theory is : iγ F (x y) = i ˆ ( S (k) + m 2 ( p 2 m 2)) e ik(x y) d 4 k (2π) 4 2m 2 (k 2 m 2 + iε) ( S 00 1 ) δ 4 (x y) + 2m 2 the last term is a contact term. Accordingly to Weinberg the correct Feynman rules are obtained by eliminating this term. Then iγ F (x y) = i ˆ ( S (k) + m 2 ( p 2 m 2)) e ik(x y) d 4 k (2π) 4 2m 2 (k 2 m 2 + iε) M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 25) de Guanajuato November Mexican 2015 Workshop25 on/ Partic 26

31 Conclusions and remarks. 1 The dirac formalism can be interpreted as a projection on to parity eigenstates in (j, 0) (0, j) 2 We have generalized this to j = 1 based on the parity based covariant basis construction. 3 The formalism yields a constraint dynamics, all constraints being second class. 4 We performed the canonical quantization following Dirac s guidelines. 5 The algebra of the S tensor is fundamental for the calculations in QFT. 6 The commutator of the elds gives a non conventional result that comes from the constraints of the theory. 7 The propagator involves not only the on shell polarization sum, but also involves terms proportional to p 2 m 2. 8 Possible extensions and applications ongoing... M. Napsuciale, S. Rodriguez, R.Ferro-Hernández, Spine S. Gomez-Ávila one elds(slide (Universidad 26) de Guanajuato November Mexican 2015 Workshop26 on/ Partic 26