3 DE LOS DATOS EN UN REACTOR DISCONTINUO

Evidently, the Weinberg-Witten theorem does not forbid photons, because they carry no conserved charge. It also does not forbid theW± and Z bosons because they are massive. But the Standard Model contains charged, massless spin-1 particles (the gluons) as well as massless spin-2 particles (the gravitons). How is this possible? The resolution of this question helps to clarify the necessity for local gauge invariance.

In a Yang-Mills theory, LYM=− 1 4F a µνFaµν+Lmatter(ψ, Dµψ) (2.81)

the gauge-invariant current

j_aµ= δSmatter

δAa µ

(2.82)

is not conserved, because it obeys the equation Dµhjaµi= 0, rather than∂µhjaµi= 0. Fur-

thermore,hjaµivanishes for one-particle gauge field states. Therefore considering the matrix

elements of thisjaµ between gauge boson states in Yang-Mills theory would avail us nothing

because the limit in Eq. (2.74) would be zero.

What we actually want is a current that measures the flow of charge in the absence of matter (i.e., for the Yang-Mills bosons alone) and that is conserved in the sense∂µhJaµi= 0:

J_aµ=−F_cµνfcabAbν , (2.83)

where the f’s are the structure constants of the gauge group. Conservation follows imme- diately from the equation of motion for Eq. (2.81). This is, in fact, the conserved current obtained through Noether’s theorem from the global gauge invariance of Eq. (2.81) with- out matter. But the current in Eq. (2.83) is obviously not gauge-invariant, because it is composed of a gauge-invariant field-strength tensor F and a non-gauge-invariant field A. Therefore, under the action of a Lorentz transformation Λ,

Jµ

a →ΛµνJaν+∂µΩa (2.84)

and it is not, consequently, Lorentz-covariant. If we tried making it Lorentz-covariant by introducing an unphysical extra polarization of the gauge boson, then the theorem would fail because the helicities would not be Lorentz-invariant, invalidating the choice-of-frame procedure used to arrive at Eq. (2.77).

To put this in another way, in a gauge theory the physical |p,±ji states are actually equivalence classes, because two states related by a gauge transformation represent the same physics. A technical way of thinking about this is that the physical states are elements of the BRST cohomology ([13]). Therefore, matrix elements such as those in Eq. (2.69) are only well-defined if the operatorjµ _{is BRST-closed, which requires the operator to be}

gauge-invariant. It is well known that Yang-Mills theories do not allow the construction of gauge-invariant conserved currents.

Einstein-Hilbert gravity

S =

Z

d4x√−g[R+Lmatter(φ,∇µφ, gµν)] , (2.85)

where the fieldφstands for all possible matter fields of any spin. The covariant stress-energy tensor Tµν = √1 −g δSmatter δgµν (2.86)

obeys∇_µhTµνi= 0 rather than∂µhTµνi= 0, andhTµνi= 0 for any state with only gravi-

tational fields. What we want is therefore notT, but rather

Θµ_ν = ∂R

∂(∂µgαβ)

(∂νgαβ)−gµνR . (2.87)

But recall that the Ricci scalar Rcontains not only the metric and its first derivatives, but also terms linear in its second derivatives. In order to define Θ we therefore need to do the usual trick of integrating by parts and setting the boundary terms to zero in order to get rid of the second derivatives in R. This means that R is no longer a covariant scalar and therefore Θ is not a covariant tensor, but rather a pseudotensor.

It is well known that gravitational energy cannot be defined in a covariant way. For instance, the energy of gravity waves on a flat background is localizable only for waves trav- eling in a single direction, which is not a coordinate-invariant condition (see, for instance, Chapter 33 in [14]). A general Lorentz transformation of the graviton fieldhµν _{will destroy}

this condition. This means that the stress-energy pseudotensor Θµν for gravitons involves a fieldhµν that does not transform like a Lorentz tensor. Its matrix elements are therefore not Lorentz-covariant. Once again, if we attempt to remedy this by introducing unphysical extra polarizations of the gravitons, the Lorentz invariance of the helicity is lost.

Otherwise stated, in a theory with diffeomorphism invariance like GR, the physical states are equivalence classes, because two states related by a coordinate transformation represent the same physics. The matrix elements in Eq. (2.69) are only well defined if the operator Tµν is BRST-closed, but GR admits no local BRST-closed operators, and thus evades the Weinberg-Witten theorem.

Notice that even in theories with a local symmetry, such as QCD or GR, the Weinberg- Witten theorem does rule massless particles of higher spin that carry a conserved charge

associated with a symmetry that commutes with the local symmetry. For instance, the au- thors of [4] point out that their result forbids QCD from having flavor non-singlet massless bound states with j ≥1, since flavor symmetries commute with the SU(3) local gauge symmetry. Similarly, a j = 1 gauge theory cannot produce composite gravitons with Lorentz-covariant spectra, because translations in flat Minkowski space-time commute with the gauge symmetry. Gauge theories admit the conserved, Lorentz-covariant Belinfante- Rosenfeld stress-energy tensor ([15]).