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M. H. Friedman, Y. Srivastava and A. Widom

arXiv:hep-ph/9206253v2 1 Sep 1992NUB 3053/92Wightman Functions in QCDM. H. Friedman, Y. Srivastava and A. WidomPhysics DepartmentNortheastern UniversityBoston, MA 02115, USAandDipartimento di Fisica & INFNUniversita di PerugiaPerugia, ItalyABSTRACTThe constraint imposed by Gauss’ law is used to show that the matrix elements of n-point WightmanFunctions of gluon field and quark current operators at different space time points vanish when taken betweenphysical states.

– 2 –1. Introduction.Christ and Lee[1,2] have warned against the common practice of starting from a formal path integralapproach prior to choosing a gauge.

In fact, for QCD they chose the (time)-axial (or temporal) gauge[3,4,5]which has a well defined Hamiltonian and then derived the Feynman rules for different gauges by appropri-ate coordinate transformations. In the temporal gauge, their construction of the path integral generatingfunctional requires for consistency that it operate only between color singlet states.For our purposes it will be convenient to work in the canonical formalism[5].

In section 2 we point outthat just implementing the constraints, due to Gauss’ law, on physical states in the temporal gauge leadsus to a pleasing result: all transition matrix elements of the color electric and magnetic fields as well as ofcolor carrying currents between physical states vanish. This forecloses the possibility of any physical statewith non-zero color appearing.

In section 3 we show that two point Wightman functions of quark currentand gluon fields in physical states vanish when the two points do not coincide, and in section 4 we extendthe results to n-point functions.2. The Physical States.In the temporal gauge, Aa0 = 0, the color electric fieldEai (x, t) = −∂Aai (x, t)/∂t.

(1)We will state the canonical commutation relations a little more carefully than is customary, so as toavoid inconsistencies.To this end let us consider the state vector Ψ which is a member of an Hilbertspace spanned by a complete set of normalizable functions of the vector potentials Aai , obeying appropriateboundary conditions. Let F be an operator such that FΨ is also a member of this Hilbert space.

Then thecanonical equal time commutation relations are given by[Eai (x, t), F(t)] = iδF(t)δAai (x, t). (2)The Gauss’ law operator Ga(x, t)Ga(x, t) = ∂iEai (x, t) −gf abcAbi(x, t)Eci (x, t) −Ja0 (x, t),(3)constraint on physical states |γ⟩reads[1,3,4,5,6,7,8,9,10]Ga(x, t) |γ⟩= 0.

(4)Unlike the QED case, the Gauss operator Ga generates rotations in color space and as such rotates the colorelectric Eai and magnetic Bai fields as well. For example,[Ga(x, t), F bµν(x′, t)] = igf abcF cµν(x, t)δ3(x −x′).

(5)where F aµν is the color electromagnetic field tensor.Taking the matrix element of Eq. (5) between arbitrary physical states γ and γ′ and using Eq.

(4) wefind that⟨γ′|F aµν(x, t)|γ⟩= 0. (6)An identical answer ensues for the color current density Jaµ, viz.,⟨γ′|Jaµ(x, t)|γ⟩= 0.

(8)Vanishing of all of the above can be satisfied only for physical states which are color singlets[11].Theequivalence of temporal and Coulomb gauges in QCD[1,2], would then imply that an identical result holdsthere as well. Note that it is only for gauge covariant quantities that one can meaningfully speak about theirvalue.

If they become null identically in one gauge they remain so in any other gauge. In contrast, we do nothave a similar argument regarding non gauge covariant quantities such as the vector potential.

We further

– 3 –note that we have assumed there is no symmetry breaking, so that a similar argument cannot be applied toSU(2) × U(1).3. Two-Point Wightman Functions.Having deduced that the physical matrix elements of non-color singlet operators vanish, we next turn toconstraining the matrix elements of color-singlet operators which are composites of other colored operators.It is not difficult to show that these are devoid of absorptive parts.

By way of illustration consider thecommutator of the product of YM fields F bµν and F cξσ, with the Gauss law operator Ga. Using Eq.

(5) andthe fact that the Ga(x) are independent of x0, we findf abc Ga(x), F bµν(y)F cξσ(z)= 3igF aµν(y)F aξσ(z)δ3(x −y) −δ3(x −z). (9)Taking Eq.

(9) between arbitrary physical states γ and γ′ and using Eq. (4) yieldsγ′|F aµν(y)F aξσ(z)|γ= 0(10)when y ̸= z.

Employing the Lorentz invariance of the theory we then find that it is true for all yµ ̸= zµ. Asimilar argument holds for the quark current operators either by themselves or in combination with F aµν.4.

N-point Functions.We now construct operators V a(x) which are composites of current density and gluon fields at the samespace time point and transform as vectors under the gauge group (for convenience). (In general they willalso carry Lorentz indices as well, but which we now suppress).

Let us then consider the following matrixelements in physical states of the commutators< α|[Ga(x), V b1(y1)V b2(y2) · · · V bn(yn)]|β >=ig < α|[f ab1cV c(y1)V b2(y2) · · · V bn(yn)δ3(x −y1)+f ab2cV b1(y1)V c(y2) · · · V bn(yn)δ3(x −y2)+ · · · + f abncV b1(y1)V b2(y2) · · · V c(yn)δ3(x −yn)]|β >= 0(11)using Eq.(4). Thus, if yi ̸= yj for all i ̸= j then each of the coefficients of the δ3(x −yi) must vanish.

Asbefore, the y0i are arbitrary. Hence, once again, appealing to the Lorentz invariance of the theory, we findthat these coefficients must vanish for space like, time like and light like separations of all the yi.Furthermore, there is only one f abc for a given a and b.

Thus, f abcV c(y) is a single term and then-point functions must vanish unless two or more of the yi are equal. In this latter case we may attempt toextract the singlet content at such points by contracting with f abc, dabc and δab.

If after all such contractionsthere remain terms with non zero color content at one or more points yi then they become matrix elementsbelonging to a smaller n and hence they must also vanish. Matrix elements which have only singlet operatorsat different space time points need not vanish.5.

Conclusions.The physical states of QCD are ones which satisfy Gauss’ law and hence have zero color. Furthermore,Wightman functions in physical states, which involve color non-singlet operators at different space timepoints, vanish identically.

– 4 –Acknowledgements.This work was supported in part by INFN in Italy and by DOE in the United States.REFERENCES[1] T. D. Lee, ”Particle Physics and Introduction to Field Theory”, Harwood Academic Publishers, NY(1988), Chapters 18 and 19. [2] N.M. Christ and T.D.

Lee, Phys. Rev.

D22, 939 (1980). [3] J. Schwinger, Phys.

Rev. 127, 324 (1962); 130, 406 (1963).

[4] R. P. Feynman, Acta Physica Polonica 24 697 (1963). [5] H. Weyl, ”The theory of groups and Quantum Mechanics”, Dover Publications, New York, NY, firstpublished in 1950.

(It is a republication of the English translation of Gruppentheorie und Quantenmechanikoriginally published in 1931). [6] B. Sakita, ”Quantum Theory of Many-Variable Systems and Fields”, World Scientific Co. Pvt.

Ltd(1985), Singapore. [7] J. L. Gervais and B. Sakita, Phys.

Rev. D18, 453 (1978).

[8] J. Kogut and L. Susskind, Phys. Rev.

D9, 3501 (1974); D11, 395 (1975). [9] S. Mandelstam, Annals of Phys.

19, 1 (1962). [10] A. M. Polyakov, ”Gauge Fields and Strings”, Harwood Academic Publishers, NY (1987)[11] A similar argument was presented by K. Cahill in a preprint, Indiana University, IUHET 34 (1978),(unpublished).

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