Igor V. Kanatchikov Institute of Theoretical Physics, Free University Berlin Arnimallee 14, D Berlin, Germany

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1 PRECANONICAL QUANTIZATION OF YANG-MILLS FIELDS AND THE FUNCTIONAL SCHRÖDINGER REPRESENTATION Igor V. Kanatchikov Institute of Theoretical Physics, Free University Berlin Arnimallee 14, D Berlin, Germany December 09, 2002 Abstract Precanonical quantization of pure Yang-Mills fields and its relation with the functional Schrödinger representation in the temporal gauge are discussed. Key words: Yang-Mills theory, De Donder-Weyl formalism, precanonical quantization, functional Schrödinger representation, mass gap. 1 Introduction The problem of quantum Yang-Mills (YM) theory has been extensively discussed for more than three decades resulting in such breakthroughs as the asymptotic freedom, the renormalizability proof, the Faddeev-Popov technique, the BRST symmetry, the Seiberg-Witten theory, duality and others. However, some of the fundamental issues such as the confinement problem and the existences of the mass gap are still not understood. It is therefore feasible to explore the potential of new approaches in solving these problems. A new method of precanonical quantization of fields which is based on a manifestly covariant (space-time symmetric) version of the Hamiltonian formalism [1 9] has been proposed recently in our papers [10 14]. It is conceptually different from the standard picture of quantum field theory as an infinite dimensional quantum mechanics. Instead, precanonical quantization is based on the picture of classical fields as multi-parameter generalized Hamiltonian systems in the sense that all space-time variables enter on the equal footing as analogues of the single time variable in Hamiltonian mechanics. This guarantees the manifest covariance of the formulation. The corresponding (precanonical) configuration space is a bundle of field variables over the space-time; the classical field configurations are the sections of this bundle. The wave functions of quantum fields, not functionals, live on this bundle. Quantization of the abovementioned generalized Hamiltonian systems representing field theories leads to a multi-parameter, Clifford-algebraic On leave from Tallinn Technical University, Tallinn, Estonia 1

2 generalization of quantum theory which reduces to the familiar complex Hilbert space quantum mechanics in the case of (0+1) dimensional space-time whose corresponding Clifford algebra is the algebra of the complex numbers. The manifest covariance and the finite dimensionality of the analogue of the configuration space (on which the wave functions live) are the obvious advantages of the precanonical approach to field quantization. Unfortunately, many aspects of the relation of this approach to the standard techniques and notions of quantum field theory are not explored yet. However, a remarkable connection between precanonical quantization and the functional Schrödinger representation is already established [15]. In this paper our aim is to consider precanonical quantization of pure YM theory and to demonstrate its connection with the functional Schrödinger representation of quantum YM theory. As a by-product, the precanonical approach is used to argue the existence of the mass gap in the pure YM theory. 2 The De Donder-Weyl Hamiltonian formulation of YM theory. The Lagrangian density of pure Yang-Mills theory is given by Following the De Donder-Weyl (DW) Hamiltonian formulation [1 6,9] we define the polymomenta The antisymmetrization in the left hand side of the second equation makes the DW Hamiltonian equations consistent with the primary constraint 3 Precanonical quantization of YM theory According to the prescriptions of precanonical quantization [10 15] we set It should be noticed that the DW Hamiltonian operator of YM theory (3) is not scalar as in the case of the scalar field theory [10, 3, 12, 15]. It can be decomposed into two parts: To see it let us write the covariant Schrödinger equation for the simple wave function assuming ψ µν = 0. Here we set for simplicity κ = 1. From (4) it follows i µ ψ µ = H 0 ψ + H µ ψ µ, (3.1) i µ ψ = H 0 ψ µ + H µ ψ, (3.2) i [µ ψ ν] = H [µ ψ ν]. (3.3) 2

3 If H µ = 0 and there are no external fields, i.e. µ H = 0, then (8) is the integrability condition of (7) and the restriction to a simple wave function = ψ + ψ ν γ ν is possible. If H µ 0and µ H = 0 the integrability condition of (7) takes the form i.e. H 0 i [µ ψ ν] H [µ H 0 ψ ν] H [µ H ν] ψ =0, 4 A relation with the functional Schrödinger representation In this section we explore how the seemingly unusual precanonical quantization of YM theory is related to the familiar canonical quantization in the functional Schrödinger representation. 4.1 Canonical quantization of pure YM theory. A reminder. Let us briefly recall the functional Schrödinger representation of the YM theory in the temporal gauge A a 0 (x) = 0 [16 21]. The canonical momenta are given by The canonical momenta are represented (in the A-representation) by the operators The necessity for regularization arises here because of the second variational derivative at equal points. We introduce a point-splitting based on a regulator K ɛ (x, x ) satisfying lim K ɛ(x, x )=δ(x x ). ɛ 0 Therefore, the regularized functional Laplacian in (4) takes the form In addition to the Schrödinger equation the wave functional fulfills the Gauss law constraint on the physical states: 4.2 The derivation of the functional Schrödinger representation from the precanonical approach It is definitely of interest to understand how the functional Schrödinger representation of quantum YM theory can be related with precanonical quantization. In what follows we will show that the functional differential Schrödinger equation can be derived from the covariant Schrödinger equation of the precanonical approach. The relation of this kind has been studied in the case of scalar field theory in [15]. Its extension to the pure YM fields in the temporal gauge is presented below. 3

4 The basic idea is that the Schrödinger wave functional ([A(x)],t) which represents the probability amplitude of simultaneously observing the field configuration A = A(x) at the moment of time t can be seen as a joint probability amplitude of simultaneously observing the respective values A(x) atthespatialpointsx (at the moment of time t). Those amplitudes are given by the wave function (A, x, t) restricted to the Cauchy surface : (A a µ = Aa µ (x),t = const). Using the result of [15], in the temporal gauge (x) = 0 the corresponding composed amplitude is written as A a 0 Let us show that using the Ansatz (4.7) we can derive the familiar functional differential Schrödinger equation of the YM theory from the covariant Schrödinger equation (3.4). For this aid let us find the Schrödinger-type equation fulfilled by the functional amplitude (7) taking into account the covariant Schrödinger equation obeyed by. From (7) we obtain { i t = tr { δ δa i a (x) = tr κ { ( δ 2 δa a i (x)δaa i (x) = tr κ 2 1 κ 1 A i a dx 1 i t }, (4.1) A i a 1 }, (4.2) A i a )} κδ n 1 (0) 1 A i 1 a A i + κδ n 1 (0) 1 2 a A i. (4.3) a Ai a The singularity δ n 1 (0) (n is the space-time dimension) arises from the second functional differentiation at equal points. The simplest regularization K ɛ (x, x ):= { 1/ɛ n 1 if x x ɛ, 0 if x x >ɛ amounts to the replacement of δ(0) with the momentum space cutoff 1/ɛ. The latter has its counterpart in precanonical quantization as the constant κ which has the meaning of 1/ɛ n 1. Under the regularization δ n 1 (0) κ we obtain Now, let us substitute into (8) the expression of i t which one obtains from the wave equation on the restricted wave function. The letter is derived from the covariant Schrödinger equation (3.4): in the temporal gauge (i.e. by just dropping out A a 0)weobtain ( d i t = iα i dx [ia a i j] (x) ) A a + βĥ, (4.4) j where d dx := i i + i A a j(x) A a j 4

5 is the total spatial derivative, and Thus, by substituting i t from (12) to (8), discarding the total divergence term dx 1 αi d dx i, and using (11), (13) we obtain In order to understand the relation of the matrix Hamiltonian Ĥ with the Schrödinger picture Hamiltonian (4) let us consider a unitary transformation of Ĥ The condition that the transformed Hamiltonian Ĥ contains no terms with the first order functional derivatives is Therefore, with the aid of the unitary transformation (16), where N is a solution of (18), the matrix-valued Hamiltonian Ĥ is transformed to the Schrödinger picture Hamiltonian operator of the YM theory: Now, let recall that by dropping out A a 0 in (4.12) we actually lost the information of the ν = 0 component of the DW Hamiltonian equations (2.5), which is the Gauss law. In order to restore it we have to require that S is invariant under the gauge transformations of A a i (x). As we have already noticed, this automatically leads to the Gauss law constraint (4.6). Thus, the total set of equations of the functional Schrödinger representation of pure YM theory in the temporal gauge is derived from the precanonical approach. Let us note that eq. (18) is formal because the second functional differentiation in (15) is formal and requires regularization. A regulator function K ɛ (x, x ) will appear then in the right hand side of (18) as dx K ɛ (x, x ) δn/δa(x ), thus making the transformation operator N explicitly depending on the regulator. In the unregularized case K ɛ (x, x )=δ(x x ) there are no solutions to (18) in the class of single-valued functionals of one argument A a i (x).1 5 On the spectrum of pure YM theory The spectrum of the DW Hamiltonian (3.3) is not easy to analyze because of the matrix interaction term CA A A. In this section we show that there exists a unitary transformation of Ĥ: 1 In [15] this issue was formally treated in the case of scalar field theory by resorting to the solution for N in terms of a functional of two arguments. The present treatment which recalls the necessity of regularization seems to be more satisfactory. 5

6 For the transformed DW Hamiltonian we obtain (in the symbolic notation, for short) Ĥ = 2 κ 2 2 AA + 2 κ 2 2 AA(iN)+ 2 κ 2 2 A(iN) A (in)+ 2 κ 2 A (in) A e i ˆN ( ig κ 2 CA A) A(iN)e i ˆN e i ˆN ( ig κ 2 CA A)ei ˆN A. (5.1) If we wish to transform away the A terms from the DW Hamiltonian, the operator ˆN has to fulfill the equation Note, however, that the right hand side of (3) is not a gradient. solution of (3) should be understood as a functional: Therefore, the In the temporal gauge A a 0 = 0 the transformed DW Hamiltonian assumes the form It is interesting to note that the Hamiltonian Ĥ admits a factorization In conclusion, let us note that the precanonical framework seems to represent a mathematically better defined foundation of field quantization than canonical or path integral quantization which involve not rigorously defined mathematical notions and require arbitrary regularizations. We have essentially demonstrated that the (unregularized) functional Schrödinger representation resulting from canonical quantization is a singular limit κ δ n 1 (0) of precanonical quantization. References [1] M.J. Gotay, J. Isenberg and J. Marsden, Momentum maps and classical relativistic fields, Part I: Covariant field theory, physics/ (and the references therein). [2] I.V. Kanatchikov, Canonical structure of classical field theory in the polymomentum phase space, Rep. Math. Phys. 41 (1998) 49-90, hep-th/ [3] I.V. Kanatchikov, On field theoretic generalizations of a Poisson algebra, Rep. Math. Phys. 40 (1997) , hep-th/ [4] M. Forger, C. Paufler, H. Römer, The Poisson bracket for Poisson forms in multisymplectic field theory, hep-th/ [5] A. Echeverría-Enríquez, M.C. Muñoz-Lecanda and N. Roman-Roy, Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys. 41 (2000) , math-ph/ [6] G. Giachetta, L. Mangiarotti and G. Sardanashvily, New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific, Singapore

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