Gauge Groups in Local Field Theory and Superselection Rules

GAUGE GROUPS IN LOCAL FIELD THEORY AND SUPERSELECTION RULES

F. Strocchi

Scuola Normale Superiore, Pisa, Italy

The basic difference between conservation laws arising from invariance under a finite Lie group and those arising from invariance under an infinite Lie Group or gauge group is discussed. Conserved currents associated to local gauge groups have the property that when their chazges are conserved they define superselection rules. This theorem has strong physical implications on the electric charge superselection rule, the presumed baryon and lepton superselection rules, the existence of (hidden) color variables in non-abelian gauge theories and on possible explanations of the non-observability of quarks. 88

In general a symmetry of the Lagrangian gives rise to a conservation law. When the symmetry group is a continuous (Lie) group one has the additional property that the conserved quantity Q is generated by a local current J (x) : Q = J|Jo(X) d3x. The deep implications of this property in quantum field theory have been extensively discussed [i] in the literature (local symmetries) . A natural question is then: is there any difference between a conservation law arising from a symmetry of the theory under a finite Lie group and one arising from a symmetry under an infinite Lie group or gauge group? In going from a finite Lie group to its associated local gauge group one does not gain any new conservation laws, one gets however an additional restric- tion on local current generating the conserved quantity Q .

Proposition I [2]. The invariance of the Lagrangian under an infinite

Lie group or gauge group implies the existence of a divergenceless current J (x) generating a conserved quantity Q . Moreover J (x) is of the form

J (x) = 3 V G ~ (x) (I) where G (x) is antisymmetric tensor, ~w The above result has deep implications in the quantum field theory formulation

Proposition 2 [3]. In a local quantum field theory conserved currents arising from local gauge invariance give rise to superselection rules if the gauge invariance is not broken. Therefore a conserved quantity

Q can be associated to a local gauge invariance of the Lagrangian only if Q is superselected.

Proof. The idea of the proof is simple. It exploits the locality of the observables, a property which will not be discussed here since its connection with Einstein's causality is well known [4], together with eq. (I). For any local observable A one has

[Q,A] = lim 1 [J (x),A] d~x = lim [I [3 i G . (x),A]d 3x = 0 R÷~ j o R÷~ J oi Exl

The above Prop.2 provides in fact a rigorous proof of the superselection rule of the electric charge; in this case the content of Prop. I is the validity of the Maxwell's equations (again for a careful di- scussion of the gauge problem see ref.[3]). Prop.2 provides also non trivial informations about the Yang-

Mills color gauge theory suggested by Fritzsch, Gell-Mann and Leut- wyler [5]. The introduction of color is forced by one the main problem related to quarks, that of quark statistics. The basic assumptions of FGML scheme are: i) the theory is of the Yang-Mills type and the color SU(3) is unbroken ii) observables are color singlets iii) physical states are color singlets. Prop.2 proves that ii) follows from i). For a detailed proof one may follow the pattern discussed in Ref.[3]. Finally we want to discuss property iii) . This is related to the problem of reconciling the success of the quark model and the failure of detecting quarks. Recently much attention has been paid in finding a mechanism for keeping the quarks bounded into systems of zero tria- lity (quark confinement). This philosophy implies that, e.g. the q-q potential does not decrease at infinity sufficiently fast to allow quarks to escape from the bounded system [6]. It seems difficult to understand the above mechanism in a local field theory, since the decrease of the potential is associated to the cluster property and in a local field theory satisfying Wightman axioms the slowest decrease at infinity is like I/r , [7]. We will see that the non abelian character of the gauge group allows to escape the above difficulty and that, jus a~n the two dimensional QED, the cluster property may fail if an infrared mechanism is acting. To show that this may happen without vio- lating locality and that the q-q potential may behave at infinity N like r , N non negative integer, one must spell out the basic pro- 90 perties of a local field theory of the Yang-Mills type. Definition. An indefinite metric quantum field theory is a theory speci lied by : a) a set of fields {~i } defined as operator valued tempered distributions on a Hilbert space H , and having a common dense domain D ; b) a bounded hermitean sesquilinear form <.,-> on H , called the metric: <-,-> = (.,~-) ; c) a representation U of the Poincar~ + group in D , unitary with respect to <-,.> : U nU = ~ ; d) a disting guished space H' C H such that <-,-> is non negative on H' and the unique translationally invariant vector ~ , the vacuum, belongs to H' O and it is cyclic with respect to the set of fields {#i } ; e) (spectral condition) the Fourier transform of the Wightman functions have support contained in V + ; f) (locality) for (x-y) 2 < 0 one or the other of [#i(x), ~j (y)]+ = 0 holds. A local theory of the Yang-Mills type is expected to be a local quantum field theory is the sense of the Definition above.

Proposition 3. (Cluster property) In a quantum field theory of the type of Definition above, with a mass gap (0,M) the cluster decomposition for the Wightman functions holds

W(x I ..... xj,xj+1+la .... Xn+ha) - W(x 1,...xj)W(xj+ I .... Xn ) ÷ 0 when the real positive number I ÷ ~ and a is a spacelike vector, the convergence being in S' . Moreover if B1(Xl), B2(x 2) denote two clu- sters f B i ix)=fdx± j dx (il fi(x .... x r(i) l Ix ÷xil Ix ÷x i) f.e~,1 i = 1,2 , one has, = Xl-X2 '

I - <~o'Bl (xl)To > I

C[~] -~ e-M[~]l~] 2N(

Proof. The crucial point is to fully exploit the spectral condition and the locality property. One defines as in Ref.7 91

hlz(~) H <~0'Bz (xl) B2(xz)~o > - <~O'B1 (xg~0><~0'Bz(x2)~0 >

[8] and similarly h21. Then one may show that h12(p) can be written

in the form h12(p) = []NP F(p), where F(p) is a bounded complex measure with the same support properties as h12(P). The same holds for N' h21 (P) = [] F'(p) and by locality and edge of the wedge theorem one P finds N = N' , F(p) = F' (p). The Fourier transform H(~) of F(p) then satisfies all the properties of the standard cluster decomposition function [7] Since h(~) = ~2N H(~) , the theorem can be easily prov- [7] ed by following the standard argument

The above Proposition shows that a qq potential decreasing at more slowly than I/r is incompatible with locality unless an infrared mechanism is present (infrared slavery).

Proposition 4 In the case of no mass gap one has

I + [~o] l where N is a suitable non negative integer. So that the cluster property may fail.

Similarly statements can be proved for the spectral representation of the two point function (the analogue of the Lehman spectral representation for Yang-Mills theories) or for the JLD-representation .

Proposition 5 The failure of the cluster property is consistent with locality if the theory is of the FGML type.

Proof. It is not difficult to show that in such a theory locality re- [3] quires the indefinite metric . Therefore the space time translations

U(a) need not to commute with the metric ~ All is required in order to have translationally invariant Wightman functions is that U(a) are

H-unitary. Thus one falls in the class of theories of the Definition above. For any two local states the Fourier transform of <~,U(a)~> need not to be a bounded fast decreasing measure as in the case of unitary U(a)'s . The exploitation of locality leads to the results of Prop. 4.

It is important to stress the r~le of the non abelian character 92 of the gauge group. This can be easily seen, e.g. in the two point function of the vector field

<~0'A~ (x) Ab(y)~ ~0 > = gpv Fab(x) + apaw Gab(x)

In the abelian case F = a A - a A is an observable and there- ~ ~ w w p fore the Fourier transform of Fab(x) must be a measure, so that the possible non unitary of the translations may affect only the unphys- ical purely gauge part ab(p) In the non abelian case F ab and G ab ~v ~v are not observable and there is no condition of ~#ab and ~d ab which may both fail to be measures. This in fact seems to be the case, as perturbation theory suggests, due to the presence of Faddeev-Popov ghosts. In the ghost free gauges, there are problems with positivity, an indication that again ~ab and ~ab cannot be measures. Another basic difference between the abelian and the non abelian case is that in the first case the local charge, i.e. the charge [ jo QR = I (x) d3x confined in a bounded region, is observable, J Ixl

REFERENCES

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