Tipo de muestreo para el tamaño de muestra en la ENVIPE

The variation of the electromagnetically interacting Dirac matter action, ignoring the contribution of the Maxwell field itself for now, is given by

δS= Z

d4x δ√−gL +√−gδL, (5.42) where the Lagrangian is given by (5.32), but with appropriate modifications to make it covariant in curved space. Since the invariant volume element d4x and

√

−gare scalar densities of weight−1 and +1 respectively, we must arrange for the Lagrangian to be manifestly a scalar. Notice that (5.32) contains a term dependent on the Levi-Civita symbol with upstairs indices, which is of weight−1. This implies that we should make the replacement

ρσκτ → √1 −g

ρσκτ_{. (5.43)}

In order to deal with the bilinear four-vectors we must introduce the vierbein fields [48], which locally relate the curved metric to the flat one

gµν =eµaeνbηab, (5.44) where Greek and Latin indices label curved and flat spacetime components respec- tively. The gamma matrices are modified such that

5.2. MAXWELL-DIRAC STRESS-ENERGY TENSOR VIA GENERAL

RELATIVITY 59

so the bilinears are now

jµ=eµaja; ja=ψγaψ, (5.46) kµ=eµaka; ka=ψγ5γaψ. (5.47) The variation of the square root of the negative metric determinant is [44]

δ√−g=−1

2

√

−g gµνδgµν. (5.48) Noting thath=√−g, wherehis the vierbein determinant, we can use the variation of (5.44) to alternatively write this as

δh=−heµa(δeµa), (5.49) implying the reciprocal variation

δ(h−1) =h−1eµa(δeµa). (5.50) In curved space, the Levi-Civita term in (5.32) becomes

h−1ρσκτ(∂ρeσa)jajκkτ+h−1ρσκτeσa(∂ρja)jκkτ. (5.51) Introducing the covariant derivative causes the first term to vanish, due to the tetrad postulate [48]

∇µeνa= 0. (5.52)

Expanding out all of the vierbein fields in the second term, we find that

ρσκτeσa(∂ρja)jκkτ =abcd(∂ajb)jckd, (5.53) which implies that for any curved coordinate components, this term is always equal to the flat spacetime version, so it is automatically covariant. We find that the covariant bilinear electromagnetically interacting Dirac matter Lagrangian has the form L = 1 2(σ 2_−ω2₎−1_[igρσ_e σaka(ω∂ρσ−σ∂ρω) +h−1ρσκτeσaeκbeτc(∂ρja)jbkc] −mσ−qgρσeσajaBρ. (5.54)

The variation with respect to deformation of the vierbein field is δL = 1

2(σ

2_−ω2₎−1_{iδ(gρσ_e

σa)ka(ω∂ρσ−σ∂ρω) +ρσκτδ(h−1eσaeκbeτc)(∂ρja)jbkc}

−qδ(gρσeσa)jaBρ. (5.55)

From the variation of (5.44), we find that

δ(gρσeσa) = 2(δeρa) + (δeσb)eρbeσa. (5.56) Using the fundamental vierbein property

5.2. MAXWELL-DIRAC STRESS-ENERGY TENSOR VIA GENERAL

RELATIVITY 60

eµa(δeµb) =−(δeµa)eµb, (5.58) we find that the first and last terms in (5.55) are

iδ(gρσeσa)ka(ω∂ρσ−σ∂ρω) =−ikµ(ω∂aσ−σ∂aω)(δeµa), (5.59)

−qδ(gρσeσa)jaBρ=qjµBa(δeµa). (5.60) Following a similar process, we find that the second variational term is

ρσκτδ(h−1eσaeκbeτc)(∂ρja)jbkc

=h−1[−eµaρσκτeσd(∂ρjd)jκkτ +ρµσκ(∂ρja)jσkκ+ρσµκeσb(∂ρjb)jakκ +ρσκµeσb(∂ρjb)jκka](δeµa). (5.61) Gathering the deformed terms together, we can write the variation of the Lagrangian as δL = 1 2(σ 2_−ω2₎−1_{−ik µ(ω∂aσ−σ∂aω) +h−1[−eµaρσκτeσd(∂ρjd)jκkτ +ρµσκ(∂ρja)jσkκ+ρσµκeσb(∂ρjb)jakκ+ρσκµeσb(∂ρjb)jκka]}+qjµBa (δeµa), (5.62) with the associated action variation being

δSD= Z d4x√−g −eµa 1 2(σ 2_−ω2₎−1_[ikρ_(ω∂ ρσ−σ∂ρω)+ρσκτ(∂ρjσ)jκkτ]−mσ−qjρBρ +1 2(σ 2_−ω2₎−1_{−ik µ(ω∂aσ−σ∂aω) +h−1[−eµaρσκτeσd(∂ρjd)jκkτ +ρµσκ(∂ρja)jσkκ +ρσµκeσb(∂ρjb)jakκ+ρσκµeσb(∂ρjb)jκka]}+qjµBa (δeµa). (5.63) From the general form of the action variation (5.35), a relationship between (5.63) and the stress-energy tensor can be obtained [48]

δSD= Z d4x√−g uλaδeλa= 1 2 Z d4x√−g Tµνδgµν, (5.64) which implies that

uµa= 1 2(Tµλe

λa_+T

λµeλa). (5.65)

RecognizingTµν as symmetric gives Tµν =

1 2(eµauν

a_+e

νauµa). (5.66) Identifying the contents of the external parentheses in (5.63) with uµa, we obtain for the stress energy tensor

Tµν = −ηµν 1 2(σ 2_−ω2₎−1_[ikρ_(ω∂ ρσ−σ∂ρω) +ρσκτ(∂ρjσ)jκkτ]−mσ−qjρBρ +1 4(σ 2_−ω2₎−1_{−i[k µ(ω∂νσ−σ∂νω) +kν(ω∂µσ−σ∂µω)]−2ηµνρσκτ(∂ρjσ)jκkτ

5.2. MAXWELL-DIRAC STRESS-ENERGY TENSOR VIA GENERAL RELATIVITY 61 +ρµσκ(∂ρjν)jσkκ+ρσµκ(∂ρjσ)jνkκ+ρσκµ(∂ρjσ)jκkν+ρνσκ(∂ρjµ)jσkκ +ρσνκ(∂ρjσ)jµkκ+ρσκν(∂ρjσ)jκkµ}+ q 2(jµBν +jνBµ), (5.67) where we have evaluated at flat spacetime. This is manifestly symmetric, but it requires some additional manipulation before it more closely resembles the Belin- fante form (5.29). Consider the U(1) gauge covariant Dirac equation and its Dirac conjugate

iγσ(∂σψ)−qγσAσψ−mψ= 0, (5.68) i(∂σψ)γσ+qψγσAσ+mψ= 0. (5.69) Left and right multiplying these equations byψandψrespectively, then subtracting the second from the first and rearranging, gives

mσ= i 2 ψγσ(∂σψ)−(∂σψ)γσψ −qjσAσ (5.70) Applying the Fierz identity (5.31) and the Bµ definition (5.28), this becomes

−mσ= −1

2(σ

2_−ω2₎−1_[ikρ_(ω∂

ρσ−σ∂ρω) +ρσκτ(∂ρjσ)jκkτ] +qjρBρ, (5.71) causing theηµν dependent term in (5.67) to vanish. Now consider the combinatorial identity2

−µρσκ(∂νjρ)jσkκ−νρσκ(∂µjρ)jσkκ

=−2ηµνρσκτ(∂ρjσ)jκkτ +ρµσκ(∂ρjν)jσkκ+ρσµκ(∂ρjσ)jνkκ+ρσκµ(∂ρjσ)jκkν +ρνσκ(∂ρjµ)jσkκ+ρσνκ(∂ρjσ)jµkκ+ρσκν(∂ρjσ)jκkµ, (5.72) which can be used to obtain the final form of the variational stress-energy tensor for Dirac matter

Tµν,D= 1 4(σ 2_−ω2₎−1_{{−i[kµ(ω∂} νσ−σ∂νω)+kν(ω∂µσ−σ∂µω)]−jσkκ[νρσκ(∂µjρ)+µρσκ(∂νjρ)]} +q 2(jµBν+jνBµ). (5.73)

Comparing with the Belinfante tensor (5.29), we find that they agree

Tµν,MD = Θµν,MD, (5.74) when the gauge field stress-energy (5.26) is included on the left-hand side.

2_{This follows from the 5 term cyclic identityV}αρσκτ_+Vταρσκ_{+· · ·= 0 which holds for the}

Levi-Civita tensor multiplied by any contravariant vector quantity. With the role of Vα played

by the Kronecker δβα (for fixedβ), this yields (5.72) after contracting withηµαδνβ(∂ρjσ)jκkτ, and