QCD Breaks Lorentz Invariance and Colour

QCD Breaks Lorentz Invariance and Colour A. P. Balachandran ∗ Physics Department, Syracuse University, Syracuse, New York 13244-1130, U.S.A. Abstract In a previous work [1], we have argued that the algebra of non-abelian superselection rules is spontaneously broken to its maximal abelian subalgebra, that is, the algebra generated by its completing commuting set (the two Casimirs and a basis of its Cartan subalgebra). In this paper, alternative arguments confirming these results are presented. In addition, Lorentz invariance is shown to be broken in QCD, just as it is in QED. The experimental consequences of these results include fuzzy mass and spin shells of coloured particles like quarks, and decay life times which depend on the frame of observation [2– 4]. In a paper under preparation, these results are extended to the ADM Poincaré group and the local Lorentz group of frames. The renormalisation of the ADM energy by infrared gravitons is also studied and estimated. 1 Introduction Quantum field theory (QFT) is defined by the algebra of local observables and an ir- A arXiv:1509.05235v2 [hep-th] 27 Jan 2016 reducible representation (IRR) π of on a Hilbert space . In general, there are many A H inequivalent IRR’s π ,π ,... of defining its superselection sectors. 0 1 A For example, in QED, π0 can be the sector with total charge q0 = 0, while πn can be the sector with total charge qn. No observation can mix these sectors. In QED, the charge operator Q generates the abelian U(1) group. In QCD, the U(1) is replaced that the non-abelian SU(3) of colour. Superficially, the superselection algebra seems to be the group algebra CSU(3). We have previously argued [1] that in reality it is a maximal abelian subalgebra of CSU(3) generated by a complete commuting set (CCS). (See in this connection the Kadison-Singer conjecture and its recent proof [5].) We can assume the CCS to be generated by the two Casimir operators and the operators on representing H the λ3 and λ8 of the Gell-Mann matrices. The remaining independent generators of CSU(3) are spontaneously broken. ∗[email protected] 1 There exist elegant proofs of Roepstorff [6], Buchholz et. al. [2] and especially Fröhlich et. al. [3] that infrared effects in QED break Lorentz invariance. We adapt these arguments here to show that generic elements of SU(3) map an IRR π of to a distinct IRR π′ = π of A 6 . In Higgs theories, this phenomenon is well-known: when the Higgs field ϕ breaks SO(3) A to SO(2) say, as in the ’t Hooft-Polyakov model [7], SO(3) transformations which change the direction of ϕ at spatial infinity cannot be represented as unitary operators on . In the H same way, here, any operator which disturbs the eigenvalues of the CCS is spontaneously broken. In section 2, we review the QED result on Lorentz violation. This is then generalised to QCD in sections 3 and 4. The results of this paper can be adapted to any non-abelian gauge group. 2 The Case of QED QED is classified by a continuous family of superselection sectors. The first is its classification by the charge q . In the representation π of , the charge n n A operator Q has the eigenvalue qn: π : Q n,P, = q n,P, (2.1) n | ·i n| ·i µ where P = (P ) denotes the total momenta. Local observables cannot change qn. Then, there are the sectors with “in” state vectors [4] 3 − + + − eqn R d x[Ai (x)ωi (x)−Ai (x)ωi (x)] n,P, := n,P,ω, , (2.2) | ·i | ·i n,P, 0, n,P, , q = 0 (2.3) | ·i ≡ | ·i n 6 ± created by the infrared photons. Here Ai are the positive and negative frequency parts of + − + the electromagnetic potential in the Coulomb gauge, and the functions ωi , ωi =ω ¯i are transverse: ± ∂iωi (x) = 0, (2.4) Also they do not vanish fast as we approach infinity: 2 ± lim r xî ωi(x) = 0. (2.5) r→∞ 6 + One such typical ωi has the Fourier transform + 3 i~k·~x + 1 ωˆ (k)= d x e ω (x)= (Pi P~ kˆ kî) (2.6) i Z i P k + iǫ − · · (with ǫ decreasing to zero as usual). The momentum Pµ is the total momentum of the charged system. (We have not shown the individual momenta and charges of which P and qn are composed as they are not importatnt for our considerations.) The important point + here is thatω î is not square-integrable: ∞ 3 d k + 2 ω,ω := lim ωî (k) = . (2.7) h i |~k|→0 Z|~k| 2 ~k | | ∞ | | 2 It is then a theorem [6] that the representation of built on (2.2) is superselected: it is not A the Fock space representation. The appendix gives a derivation of (2.2). We now elaborate on the physical meaning of (2.2). Consider the current µ µ 4 dz (τ) J (x)= qn dτ δ (x z(τ)) . (2.8) Z − dτ It radiates photons of momenta k := ( ~k ,~k). We are interested in infrared photons, so we | | assume that dzµ(τ) P µ P µ = , zµ(τ)= τ , m2 = P µP , m> 0, (2.9) dτ m m µ where P µ is constant. Now (2.8) generates the additional interaction 3 µ d xAµ(x)J (x). (2.10) Z It changes the “in” state to n,ω as is shown in the appendix. | ···i The following is a further important point. If the Lorentz boost 3 2 2 Ki = d xxi[E~ (x)+ B~ (x)] + matter part (2.11) Z is well-defined in n, 0; , then it diverges in the sector n,ω; , ω = 0 : Lorentz invariance | ·i | ·i 6 is broken in the latter. We can see this as follows: 3 2 2 n,ω; d xxi[E~ (x)+ B~ (x)] n,ω; h ·| Z | ·i 3 2 2 = n, 0; d xxi[(E~ ~ω) (x)+ B~ (x)] n, 0; , h ·| Z − | ·i + − ωi := ωi + ωi (2.12) 2 1 and this diverges since ~ω (x)= ( 4 ) as ~x . O |~x| | |→∞ There is an alternative approach for these considerations due to Roepstorff [6]. It is based on the Weyl algebra and the GNS construction. For free scalar fields, in four-dimensional spacetime, the Weyl algebra has elements W (f) where f is a compactly supported real W test function, f ∞. If g is another such test function, ∈C0 W (f)W (g)= W (f + g)eiσ(f,g)/2, (2.13) σ(f, g)= i d4x d4yf(x)D(x y)g(y), (2.14) Z − d3P 1 D(x y) = commutator function = [e−iP ·(x−y) eiP ·(x−y)]. (2.15) − Z (2π)3 2 P − | 0| Since ( + m2)D = 0 if the field has mass m, σ vanishes on any function f of the form ( + m2)h, with h ∞. On quotenting out such functions, σ becomes a symplectic form. ∈C0 3 Also, the function σ(f, ) defined by · σ(f, )(y)= i d4xf(x)D(x y) (2.16) · Z − fulfills the equations of motion. Let us introduce the scalar product 3 1 d k 2 2 1/2 (f, g)= f˜(k)˜g(k), k0 = (~k + m ) , (2π)3 Z 2 k | | | 0| where f˜(k)= d4x e−ik·xf(x), g˜(k)= d4x e−ik·xg(x). (2.17) Z Z Then the Fock representation with the corresponding Hilbert space is given by the fol- H lowing state ω0 on the Weyl algebra and the GNS construction: −(f,f)/2 ω0(W (f)) = e . (2.18) If 0 is the Fock vacuum, we have, as can be checked, | i ω (W (f)) = 0 W (f) 0 . (2.19) 0 h | | i Now if F is a linear functional on test functions, we can twist ω0 to a new state ωF : i Im F (f) ωF (W (f)) = ω0(W (f))e . (2.20) Suppose we can write F (f)= η,f , η . (2.21) h i ∈H Then, by Schwarz inequality, F (f) η f . (2.22) k k ≤ k kk k Conversely, by the Riesz-Frechet theorem [9], we can write F (f)= η,f for η iff h i ∈H F (f) < c f , c = constant. (2.23) k k k k Now if there is such an η, we can check that ω (W (f)) = 0 W (Im η)†W (f)W (Im η) 0 . (2.24) F h | | i Since W (Im η) 0 is in the Fock space, the GNS representation from ω is unitarily equiv- | i F alent to the one from ω0. Any smooth ξ ∞ which is not in also gives an F : ∈C H F (f) = (ξ,f), (2.25) 4 since f is a test function and hence compactly supported. In this case, ω (W (f)) = 0 W (Im ξ)†W (f)W (Im ξ) 0 , (2.26) F h | | i but W (Im ξ) 0 is not in the Fock space. So the representation of built on W (Im ξ) 0 is | i W | i not unitarily equivalent to the Fock representation. These considerations can be adapted to QED. Let αµ be the test function for the po- µ tential in the Lorentz gauge ∂ αµ =0. If f (α)= ∂ α ∂ α , (2.27) µν µ ν − ν µ the symplectic form (modulo the kernel) is σ: 4 4 µ σ(α1, α2)= i d x d y α (x)D(x y)α2µ(y).

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