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NONPERTURBATIVE QUANTIZATION OF NON-ABELIAN GAUGE THEORIES.
A.A. Slavnov Steklov Mathematical Institute and Moscow State University


Progress in physics was usually related to the intro duction of new symmetries. Recent examples are given by gauge theories. QED may b e formulated in the Coulomb gauge, however much more transparent formulation is presented by the quantization in a manifestly covariant gauge. YangMills theory b ecame really p opular only after its formulation in the Lorentz covariant terms and explicit pro of of its renormalizability. The gauge invariance of the Higgs mo del allows to give a manifestly renormalizable theory describing a massive gauge theory. In this talk I wish to make a propaganda for a new class of symmetries, which were intro duced in my pap er rather long ago (A.A.S., 1991), but recently were applied successfully to the nonp erturbative quantization of non-Ab elian gauge theories, construction of the infrared regularization, applicable b eyond p erturbation theory, problem of soliton exitations in Yang-Mills theory


Equivalence theorems: canonical transformations, p oint transformations = + f ( ) More general transformations: n-1 n + f ( n-1 , . . = n t t The sp ectrum is changed. What ab out . ~ ) = f ( ) t the unitarity? (1)


Path integral representation for the scattering matrix

S=

exp{i

L()dx}dµ();

limt±(x) = out,in(x)

(2)

If the change (1) do es not change the asymptotic conditions, then the only effect of such transformation is the app earance of a nontrivial jacobian a a cb ~ (3) L() L( ) = L[( )] + ¯ c b For all new excitations one should take the vacuum b oundary conditions. Unitarity?


The new Lagrangian is invariant with resp ect to the sup ertransformations a = ca L ( ) (4) ca = 0; ¯a = c a On mass shell these transformations are nilp otent and generate a conserved charge Q. In this case there exists an invariant subspace of states annihilated by Q, which has a semidefinite norm. (A.A.S.,1991). For asymptotic space this condition reduces to Q0| >as= 0 (5)

The scattering matrix is unitary in the subspace which contains only excitations of the original theory. However the theories describ ed by ~ the L and the L are different, and only exp ectation values of the gauge invariant op erators coincide. In gauge theories the transition from one gauge to another may b e considered as such a change.


A very nontrivial generalization is obtained if further shifting the fields by constants. It is of variables in the path integral as it changes fields. The unitarity of the "shifted"theory is sp ecial pro of (if p ossible) is needed.

one not the not

~ transforms the L an allowed change asymptotic of the guaranteed and a

Using this metho d one can construct a renormalizable formulation of nonab elian gauge theories free of the Grib ov ambiguity. A.A.Slavnov,JHEP,0808(2008)047; Theor.Math.Phys,161(2009)204; A.Quadri, A.A.Slavnov, JHEP, 1007 (2010); A.Quadri, A.A.Slavnov, Theor.Math.Phys, 166(2011)201


A problem of unambiguos quantization of nonab elian gauge theories b eyond p erturbation theory remains unsolved. Even in classical theory the equation DµFµ = 0 (6)

do es not determine the Cauchi problem. Gauge invariance results in existence of many solutions of this equation. To define the classical Cauchi problem and subsequently to quantize the mo del one imp oses a gauge condition, e.g. Coulomb gauge iAi = 0.


Differential gauge conditions: L(Aµ, ) = 0 Grib ov ambiguity. ~ Algebraic gauge conditions: L(Aµ, ) = 0 absence of the manifest Lorentz invariance and other problems.


Coulomb gauge i A i = 0 Ai = ( A ) i a + ig abci(Abc) = 0 i (7) This equation has nontrivial solutions fastly decreasing at spatial infinityGrib ov ambiguity. In p erturbation theory the only solution is = 0.


A remedy: new (equivalent) formulation of the Yang-Mills theory using more ghost fields. Let us consider the classical (S U (2))Lagrangian 1a a ~ L = - Fµ Fµ + (Dµ)(Dµ) - (Dµ)(Dµ) 4 +i[(Dµb)(Dµe) - (Dµe)(Dµb)]

(8)

The scalar fields (, are commuting, e, b are anticommuting) are parametrized by the Hermitean comp onents = i1 + 2 0 - i3 , 2 2 (9)


Integrating over the fields , , b, e with vacuum b oundary conditions one gets exp{i ~ Ldx}dµ = ~ exp{i Ldx}(det D2)2(det D2)-2dµ (10)

Here the measure dµ includes the gauge fixing factor and FaddeevPop ov ghosts and the measure dµ includes also differentials of the ~ fields , , b, e. The integral reduces to the usual path integral for ~ the Yang-Mills scattering matrix. The lagrangian L gives for the gauge invariant correlators the same result as the standard Yang-Mills Lagrangian: 1a a L = - Fµ Fµ (11) 4


Now we consider a different lagrangian, which may b e obtained from ~ L by the shift - g -1 m; ^ The constant field m has a form: ^ m = (0, m) ^ The new Lagrangian lo oks as follows 1a a L = - Fµ Fµ + (Dµ)(Dµ) - (Dµ)(Dµ) 4 -g -1[(Dµ) + (Dµ)](Dµm) - g -1(Dµm)[Dµ + Dµ] ^ ^ +i[(Dµb)(Dµe) - (Dµe)(Dµb)] (13) + g -1m ^ (12)

(14)

Note that b ecause of the negative sign of the kinetic term this field p osesses negative energy.This is crucial to insure the cancellation of the terms quadratic in m in the shifted Lagrangian and provide the zero mass for the Yang-Mills field.


Higgs mo del. The mo del we consider in many resp ects reminds the Higgs mo del. Instead of one scalar fields we have two scalar fields with different signs of energy and two more anticommuting scalar fields. The presence of two commuting scalar fields with different signs of energy allows to avoid the mass generation for the vector field. As in the Higgs mo del these scalar fields b ecome gauge fields, that is by the gauge transformation they are shifted by arbitrary function. In the Higgs mo del one starts with the Lagrangian L = LY M + (Dµ)(Dµ) - 2( - µ2)2 (15)

After the shift = + µ, µ = {0, µ} a, a = 1, 2, 3 b ecomes a ^ ^ gauge field:a a + µ a(x) + . . .. Unitary gauge a = 0 is algebraic, but Lorentz invariant. However this gauge is nonrenormalizable.


Is it p ossible to invent Lorentz invariant algebraic gauge for the YangMills theory in which the theory is renormalizable? The Lagrangian (14) may b e obtained from the gauge invariant Lagrangian, describing the interaction of the complex scalar doublets with the Yang-Mills field by the shift - g -1 m; ^ + g -1m ^ (16)

Hence the Lagrangian (14) is invariant with resp ect to the"shifted"gauge transformations. In particular the
transformation of the field a = - is - g g a = m a + 2 abcb c + 2 0 a - - - 2


This Lagrangian is also invariant with resp ect to the sup ersymmetry transformations -(x) = 2i b(x) e(x) = +(x) b(x) = 0 where is a constant anticommuting parameter. crucial role in the pro of of the equivalence by the Lagrangian (8) to the standard Yangthe unitarity of the scattering matrix in the only three dimensionally transversal comp onents (17)

This invariance plays a of the mo del describ ed Mills theory. It provides subspace which includes of the Yang-Mills field.


The sp ectrum: Ghost exitations: ±, b, e, longitudinal and temp oral comp onents of Aa µ Physical exitations: three dimensionally transversal comp onents of the Yang-Mills field. The sup ersymmetry of the effective action generates a conserved nilp otent charge Q. Physical states are separated by the condition Q| >ph= 0 (18)

the states separated by this condition describ e only three dimensionally transversal comp onents of the Yang-Mills field. The ghost exitations decouple, the theory is renormalizable in the usual sense.


Conclusion.
A renormalizable manifestly Lorentz invariant formulation of the nonAb elian gauge theories which allows a canonical quantization without Grib ov ambiguity (including Higgs mo del). In p erturbation theory the scattering matrix and the gauge invariant correlators coincide with the standard ones. On the basis of this approach infrared regularization of Yang-Mills theory b eyond p erturbation theory is constructed (Slavnov, 2013). In this approach soliton exitations in Yang-Mills theory seems to b e p ossible. The problem of color confinement by means of top ologically nontrivial solitons is under consideration.