Space-Time Symmetries

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1 Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a ivergence the same proceure still applies. We choose the fiels ϕ ( x ) as various representations of Lorentz group in a imensional flat space-time manifol. In an infinitesimal proper orthochronous Lorentz transformation x x = x + ω x, ω << 1, ω = ω the fiels transform as ν ν ν ν ν 1 ν B ϕ ϕ ( x ) = δ B + ω ν B ϕ Σ (.1) where Σ ν ν = Σ are the spin matrices. ϕ ( x ) transforms like a scalar uner space- time translation x x x a, = +. The action S[ ϕ] = x ( ϕ ϕ) Λ is invariant 8

2 uner the full Poincaré group of Lorentz transformation an space-time translation. Here the omain Λ is an invariant set uner the transformation..1 Noether s Theorem ( ) The action of a fiel theory escribe by a local Lagrangian ϕ ϕ, is S [ ϕ] = x ( ϕ ϕ ), Λ (.) Suppose uner an infinitesimal iffeormorphism x x x ε ε the fiel, the Lagrangian, an the action transform as = +, << 1 ϕ ϕ = ϕ + δϕ (.3a) = +δ (.3b) [ ϕ] [ ϕ ] [ ϕ] δ [ ϕ] S S = S + S (.3c) ( ) where = ϕ, ϕ, then up to the first orer in ε [ ] [ ] = x + δs ϕ δ ε Λ (.4) If x x = x + ε, ε << 1 is a symmetry of the fiel theory, then δ [ ϕ] If the choice of the omain Λ is arbitrary, then S = 0. 9

3 [ ] δs ϕ = 0 δ + ε = 0 (.5) This is a very strong conition. In the weakest situation Λ=, so δs [ ϕ] = 0 is a necessary an sufficient conition for the existence of a vector fiel V such that δ + ε = V (.6a) V = 0 infinity (.6b) Invariance of the action S [ ϕ ] = S[ ϕ ] tells us ( x ) = ϕ ( x ) ϕ ( x ),. Using the equation of motion ϕ Π = 0, where =, one fins ϕ Π δ = Π δϕ (.7) Substituting this into eqn. (.6a), we fin ( Π V ) δϕ + ε = 0 (.8) Therefore Noether s conserve current for a space-time symmetry is = δϕ + ε J x Π x x x x V x (.9) 10

4 J is arbitrary up to the aition of a ivergence-less fiel an up to a change of scale an sign. In the subsequent analysis we will use eqn. (.9) to construct the canonical conserve currents associate with the various space-time symmetries.. Poincaré Currents Since the volume element (measure) x is invariant uner a proper orthochronous ν Poincaré transformation x x = Λ ν x + a, ( et( Λ ν ) = + 1, Λ ), the action S[ ϕ ] is invariant in arbitrary omain Λ when the Lagrangian ( x ) is a Poincaré scalar. Therefore we will apply the strong conition (.5) for calculating the Poincaré currents. Canonical stress-energy tensor is the conserve current associate with the parameter space-time translation group x x x a = +. The fiels ϕ x form a basis to the various representations of the Lorentz group an transform like scalars uner translation ν δϕ x = ϕ x ϕ x = a ϕ x, a << 1 (.10) ν We rea the conserve current from eqn. (.9) as Θ ν Π ν ν ν = ϕ η, Θ = 0 (.11) 11

5 This is the well-known canonical stress-energy tensor. Conservation follows ientically by the equation of motion. The anti-symmetric part of Θ ν is given by Θ Θ = Π ϕ Π ϕ ν ν ν ν (.1) The conserve current associate with the proper orthochronous Lorentz transformation x x = x + ω x, ω << 1, ω = ω is the canonical angular ν ν ν ν ν momentum tensor. The fiels ϕ ( x ) are representations of Lorentz group which transform like 1 ν B ϕ ϕ ( x ) = δ B + ω ν B ϕ Σ (.13) Due to the group structure of the Lorentz transformation, the spin-matrices Σ satisfy the following commutation relation [9] =Σ ν ν [, ] Σ ν Σ λρ = η ρ Σ νλ η λ Σ νρ + η νλ Σ ρ η νρ Σ λ (.14) Spin-matrices for scalar, vector, an secon rank tensor fiels are given by Σ ν = 0 (.15a) Σ ν η δ ν η ν B B δ B = (.15b) Σ ν B CD ν C B ν D C B B D ν D ν B C D = η δ δ η δ δ + η δ δ η δ δ C (.15c) 1

6 These equations can be easily generalize for arbitrary rank tensor fiels. The spin- iγ mψ = 0 an matrices for Dirac bi-spinor fiels ψ ( x ) an ψ γ ( i ) m ψ x which satisfy + = 0 (here the arrow on top of the ifferential operator enotes its irection of operation), respectively, are given by ( Σ ν ) ψ 1 ψ [, ] i = γ γ = σ 4 ( Σ ν ) ψ 1 ψ [, ] ν ν i = γ γ = σ 4 ν ν (.16a) (.16b) ν ( Σ ) ψ ψ = = ( Σ ) ν 0 ψ ψ (.16c) Dirac matrices γ ν ν satisfy Cliffor algebra [ γ, γ ] = η, an ν i ν σ = [ γ γ ] +,. From now on the fiel inices, B, C,... will be suppresse an all the equations will be interprete as matrix equations with appropriate summation over the fiel inices. Then from eqn. (.13) λ ρ ρ λ λρ [ ] 1 δϕ = ϕ ϕ = ω λρ ( x ϕ x ϕ) + Σ ϕ (.17) Noether s current (.9) efines the conserve canonical angular momentum tensor as λρ x λ ρ x ρ λ λρ λρ = Θ Θ + Π Σ ϕ, = 0 (.18) which is anti-symmetric in ( λ, ρ). 13

7 The conservation law λρ = 0 oes not follow as an ientity for arbitrary Lagrangians. Therefore Lorentz invariance constrains the canonical stress-energy tensor. ν Using Θ = 0, we fin λρ λρ λρ ρλ = 0 Θ Θ = Π Σ ϕ (.19) Hence, if a translation invariant fiel theory is also Lorentz invariant, then the anti-symmetric part of the canonical stress-energy tensor has to be a total ivergence. It is interesting to notice that: Λ xθ λρ = Λ xθ ρλ if ϕ( x ) = 0 Λ. In the next chapter we will prove the weakest set of necessary an sufficient conitions for the existence of a symmetric stress-energy tensor, an we will evelop an algorithm to construct it from the canonical stress-energy tensor over a flat space-time manifol. 14