Symmetric Stress-Energy Tensor

Transcript

1 Chapter 3 Symmetric Stress-Energy ensor We noticed that Noether s conserved currents are arbitrary up to the addition of a divergence-less field. Exploiting this freedom the canonical stress-energy tensor Θ ν can be modified to a new tensor ν ν n ν n such that = 0 and d x = d xθ Mn 0 0ν (here M denotes the spatial sub-manifold of the space-time M ). he second condition n Mn guarantees that the new tensor ν defines the same physical observable (namely, energy-momentum of the field). From Green s theorem, such a modification of Θ ν ν require the existence of an anti-symmetric Belinfante [] tensor field B ( x) such that ν ν ν ν ν = Θ + B, B = B (3.) In this chapter we will discuss a necessary and sufficient conditions for the existence of a Belinfante tensor such that ν is symmetric. Our main goal here is to introduce notations, and to summarize the results necessary to make the thesis self-contained. 5

2 3. Necessary and Sufficient Conditions heorem: he anti-symmetric part of the conserved canonical stress-energy tensor is a total divergence, if and only if there exists a symmetric stress-energy tensor []. Proof of Sufficiency: Suppose Θ ν Θ = H (3.) ν ν By definition, H ν ν = H. Choose B = H + H H ( ) ν ν ν ν (3.3) his tensor have the right anti-symmetry B ν ν = B, and also ν ν ν B B = H (3.4) Applying eqns. (3.) and (3.4) in the definition (3.), we find ( ) ν ν ν ν ν ν = Θ Θ + ( B B ) = 0 (3.5) Hence given H ν one can explicitly construct a Belinfante tensor B ν such that ν = Θ + B is symmetric. ν ν 6

3 Proof of Necessity: his is in fact trivial. If there exists a symmetric ν ν ν definition (3.), ( Θ Θ ) ν ν = ( ) B B, a total divergence. then from 3. Construction of Belinfante ensor From eqn. (.9) on angular momentum conservation we have seen that a necessary condition for a translation invariant theory be Lorentz invariant is ν ( ϕ) ν ν Θ Θ = Π Σ, a total divergence. herefore as a consequence of eqn. (3.3) a full Poincaré invariant field theory always have the following Belinfante tensor which makes ν = Θ + B a symmetric stress-energy tensor ν ν B ν = ( ν + ν ν ) ϕ Π Σ Π Σ Π Σ (3.6) It is important to note that, in general, the choice of symmetric stress-energy tensors is not unique. his will be our key to the analysis in chapter 5 to construct an improved tensor, if exists, for the scale invariant field theories. here is an alternative definition of symmetric stress-energy tensor in general relativity [35]. he functional derivative of the action minimally generalized to a metric compatible Riemannian manifold M R through the correspondence relations d d ( ν g ν ( x),, gβ = 0, d x d x g ) η is defined as a symmetric stressenergy tensor in general relativity. We will show in appendix A that these two symmetric stress-energy tensors are identical in flat space-time [3-34] 7

4 g δs δg ν gβ ( x) = η β = ν (3.7) We wish to express the angular momentum in terms of symmetric stress-energy tensor. Substituting Θ ν ν ν = B in eqn. (.8), we find λ ρ ρ λ λρ λρ x x = M + F (3.8) where F λρ ( x λ B ρ x ρ B λ ) = is anti-symmetric in (, ) and in ( λ ρ),. Dropping this anti-symmetric divergence, we obtain the conventional angular momentum tensor λρ λ ρ ρ λ J = x x (3.9) It defines the same Lorentz generators as M λρ. Due to the symmetry of ν, the conservation law J λρ = 0 is now an identity. It is important to note that ν and hence J λρ are gauge independent. he action is invariant under a gauge transformation of the Lagrangian ( x) ( x) = ( x) + Z ( x) such that Z ( x) Λ = 0. he canonical stress-energy tensor transforms as Θ Θ = Θ + ( η ) ν ν ν ν ν Z Z. reating Z ( x) as an independent vector field in the Lagrangian, one finds from eqns. (.5b) and (3.6) that 8

5 ( η η ) B ν B ν = B ν ν Z ν Z which clearly shows the gauge independence of ν ν ν ν ν ν ν, namely = Θ + B = Θ + B =. 3.3 Some Examples We can now apply the definition (3.6) of Belinfante tensor with the spin-matrices (.5a,b) and (.6a,b,c) to construct the symmetric stress-energy tensor for real scalar, vector, and Dirac bi-spinor fields Real Scalar Field Scalar fields are spinless: Σ λρ λρ = 0 = Θ and J = M. hese results also ν ν ν follow from a direct calculation using the standard Lagrangian = ( ) ϕ ϕ ϕ V (3.0) Explicit calculation shows = Θ = ϕ ϕ η (3.a) ν ν ν ν ( ) ( ) λρ λρ λ ρ ρ λ λ ρ ρ λ J = M = ϕ x ϕ x ϕ + η x η x (3.b) It is important to note that the trace of ν is d = ϕ ϕ + dv ( ϕ) (3.) 9

6 Notice that, if V ( ϕ ) = 0, then the trace vanishes identically in d =. his is a consequence of general conformal invariance Vector Field he Lagrangian for a massive free U() vector field is β m = Fβ F + A A (3.3a) 6π 8π F = A A (3.3b) β β β If m 0 the gauge freedom does not exist and the Lorentz gauge condition A = 0 is an ad-hoc constraint on the vector field A ( x) due to the equation of motion β β F m A + = 0. he canonical stress-energy tensor is η 4π F A (3.4) Θ ν ν λ ν = λ From the spin-matrices (.5b), we find the Belinfante tensor for the vector field as: B = F A 4π ν ν symmetric stress-energy tensor. Applying the equation of motion, one finds the well-known ν λν ν β m F λ F Fβ F A A ν ν = + η + η A A 4π 4 4π (3.5) 0

7 Notice that the trace of ν is d β m d = F F A A π + π β (3.6) If m = 0 ( U() gauge theory of photons), the trace of the symmetric stress-energy tensor identically vanishes in d = 4. his is again a consequence of general conformal invariance Dirac Bi-Spinor Field A real scalar Lagrangian for the free Dirac field is = ψ( iγ m) ψ ψ( iγ + m) ψ (3.7) he arrow on top of the differential operator denotes its direction of operation. he canonical stress-energy tensor is i ψγ ( γ ) ψ Θ ν = ν ν (3.8) Recalling the spin-matrices (.6a,b,c), we obtain the Belinfante tensor B ([ ] [ ] [ ] ) = ψ γ, σ + γ, σ γ, σ ψ. his expression can be further ν ν ν ν

8 simplified by using the Clifford algebra of Dirac matrices [, ] [ A, B, C ] [ B,[ C, A ] ] = [[ A, B ], C ] identity [ ] ν ν γ γ = η and the + B [ ] = ψγ, σ ψ (3.9) 8 + ν ν Using the commutation relation [ γ, σ ν ] ( η γ ν η ν γ ) = equations of motion ( iγ m) ψ ψ( iγ m) = = + 0, we obtain the symmetric stressenergy tensor for Dirac bi-spinor field i and then applying the i = ψ( γ + γ γ γ ) ψ 4 ν ν ν ν ν (3.0) Straightforward calculation yields the angular momentum tensor as J i [ ] [ ] [ ] [ ] = ψγ ( x + γ x + x γ + x γ ) ψ λρ ρ λ ρ λ ρ λ ρ λ (3.) Here we have used the notation: Q def [ β ] β β = ( Q Q ). he trace of ν is = mψψ (3.) If m = 0, then it is traceless for all space-time dimensions. Massless Dirac field is general-conformal invariant in all space-time dimensions.