Hamiltonian, Path Integral and BRST Formulations of the Restricted Gauge Theory of QCD2

We study the Hamiltonian, path integral and Becchi-Rouet-Stora and Tyutin (BRST) formulations of the restricted gauge theory of QCD2 à la Cho et al. under appropriate gauge-fixing conditions.

The theory is seen to be gauge-invariant (GI) possessing a set of first-class constraints [14]. We quantize this theory under appropriate gauge-fixing conditions (GFC's) using the Hamiltonian and path integral formulations [24] [28].
However, in the usual Hamiltonian and path integral quantization [24] [25] [26] [27] [28] of a theory under some GFC's, the gauge-invariance of the theory gets broken because of the gauge-fixing. In view of this, in order to achieve the quantization of the theory such that the gauge-invariance of the theory is maintained even under gauge-fixing, we go to a more generalized procedure called the BRST quantization [29] [30] [31], [27] [28] [29] [30] [31], where the extended gauge symmetry of the theory (called the BRST symmetry) is maintained even under gauge-fixing. In fact, this also necessitates a study of the BRST quantization of the theory to achieve a kind of complete quantization of the theory.
The paper is organized as follows. In the next section, we briefly recap the basics of the so-called restricted gauge theory of QCD 2 à la Cho et al. [1]- [13]. In Section 3, we study its Hamiltonian and path integral formulations. Its BRST formulation is studied in Section 4. Finally the summary and discussion is given in Section 5.

Restricted Gauge Theory of QCD2 à la Cho et al.: A Recap of Basics
In this section, we recap the basics of the restricted gauge theory of QCD 2 à la Cho et al. [1]- [14] and others [1]- [23]. The theory makes use of the so-called "Cho-decomposition", which is, in fact, the gauge independent decomposition of the non-Abelian potential into the restricted potential and the valence potential and it helps in the clarification of the topological structure of the non-Abelian gauge theory, and it also takes care of the topological characters in the dynamics.
The non-Abelian gauge theory has rich topological structures manifested by the non-Abelian monopoles, the multiple vacua and the instantons and one needs to take into account these topological characters in the non-Abelian dynamics. Since the decomposition of the non-Abelian connection contains these topological degrees explicitly, it can naturally take care of them in the non-Abelian dynamics.
An important consequence of the decomposition is that it allows one to view QCD as the restricted gauge theory (made of the restricted potential) which is coupled to a gauge-covariant colored vector field (the valence potential). The restricted potential is defined in such a way that it allows a covariantly constant unit isovector n everywhere in space-time, which enables one to define the gauge-independent color direction everywhere in space-time and, at the same time, allows one to define the magnetic potential of the non-Abelian monopoles.
Furthermore it has the full SU(2) gauge degrees of freedom, in spite of the fact that it is restricted. Consequently, the restricted QCD made of the restricted potential describes a very interesting dual dynamics of its own, and plays a crucial role in the understanding of QCD.
On the other hand, the restricted QCD is a constrained system, due to the U. Kulshreshtha et al. presence of the topological field n which is constrained to have the unit norm. A natural way to accommodate the topological degrees into the theory is to introduce a topological field n of unit norm, and to decompose the connection into the Abelian projection part which leaves n a covariant constant and the remaining part which forms a covariant vector field: where A µ is the "electric" potential and the Abelian projection A µ is precisely the connection which leaves n invariant under the parallel transport and makes n a covariant constant: Also, under the infinitesimal gauge-transformation: This shows that A µ by itself describes an gauge degrees of freedom. The restricted potential A µ is defined by the Abelian projection and the connection space (the space of all gauge potentials) forms an affine space. Indeed the affine nature of the connection space guarantees that one can describe an arbitrary potential simply by adding a gauge-covariant piece X µ to the restricted potential.
The above mentioned decomposition is known as the Cho-decomposition or the Cho-Faddeev-Niemi decomposition. It was introduced [2]- [23] in an attempt to demonstrate the monopole condensation in QCD. The decomposition itself and the importance of this decomposition in clarifying the non-Abelian dynamics in QCD 2 has been studied by many authors and for further details we refer to the work of Refs. [2]- [23].
The restricted potential A µ actually has a dual structure and the field strength made of the restricted potential is decomposed as: where C µ is the "magnetic" potential. Further, following the work of Refs.
[2]- [23], it is possible to introduce the magnetic potential as above (at least locally section-wise) in view of the following identity: which allows one to identify the non-Abelian monopole potential by: 1ˆn n g µ µ = − × ∂ C (6) in terms of which the magnetic field is expressed as 1ˆˆĝ n n H n g With the above connection (-albeit decomposition) one has: (8) and for the Yang-Mills Lagrangian density one has: [1]- [15]: where λ and µ λ are the Lagrange multiplier fields and The theory defined by the above Lagrangian density has a full ( )

Hamiltonian and Path Integral Formulations
We now study the Hamiltonian and path integral quantization of the above restricted gauge theory of QCD 2 (made of the Abelian projection without µ X ) defined by the Lagrangian density [2]: In the instant-form (IF) of dynamics, the above Lagrangian density reads: Here Φ is another topological scalar field constructed out of the topological field n (of unit norm) and its space derivative ( 1n ∂ ) as well as its time derivative ( 0n ∂ ), as defined above. The canonical momenta obtained for the above theory are:  1 A , and n respectively. The above equations however, imply that the theory possesses three primary constraints: The symbol ≈ here denotes a weak equality in the sense of Dirac [24]. The canonical Hamiltonian density of the theory c  is: After including the primary constraint 1 , w x t which is treated as dynamical, the total Hamiltonian density of the theory T  could be written as: The Hamilton's equations of motion of the theory that preserve the constraints of the theory in the course of time could be obtained from the total Hamiltonian: The preservation of 1 ψ for all times does not lead to any secondary constraint. The preservation of 2 χ and 2 Ω also does not lead to any new con-straints. The theory is therefore seen to possess a set of five constraints: ( ) Matrix R αβ of the Poisson brackets of the above constraints i ρ among themselves is clearly singular implying that the set of these constraints i ρ is first-class and that the theory under consideration is GI. In fact, the theory is seen to be invariant under the local vector gauge transformations: where gauge parameter ( ) is an arbitrary function of its arguments. The components of the vector gauge current ( ) The theory is clearly gauge-invariant and could now be quantized under appropriate gauge-fixing conditions (GFC's), which could e.g. be chosen as (which by no means is an unique choice): It may be important to mention here that any set of GFC's could be chosen here such that the resulting set of constraints of the theory (including the set of GFC's) becomes a set of second-class constraints so that the matrix of the total set of constraints becomes non-singular and consequently could be inverted.
The total set of constraints of the theory under these GFC's then becomes: The non-vanishing equal-time Dirac brackets (DB's) of the theory are [24]: Here one needs to remember that while making a transition from equal-time This completes the Hamiltonian and path integral formulations of the theory.
The BRST formulation of the theory is considered in the next section.

BRST Formulation
In the following, we study the BRST formulation of the theory.
with the property 2 0 δ = . We now define a BRST-invariant function of the dynamical phase space variables of the theory to be a function f such that ˆ0 f δ = .
Now the BRST gauge-fixed quantum Lagrangian density BRST  for the theory could be obtained by adding to the first-order Lagrangian density IO  , a trivial BRST-invariant function (e.g.) as follows: The last term in the above equation is the extra BRST-invariant gauge-fixing term. After one integration by parts, the above equation could now be written as: The last term in the above equation is the BRST-invariant gauge-fixing term.
Proceeding classically, the Euler Lagrange equation for b reads: which in turn (with the requirement ˆ0 b δ = ) then implies: The above equation is also an Euler-Lagrange equation (ELE) obtained by the variation of BRST  with respect to c . We define the bosonic momenta in the usual manner: but for the fermionic momenta with directional derivatives we set implying that the variable canonically conjugate to c is ( 0 c ∂ ) and the variable conjugate to c is ( 0 c ∂ ). For writing the quantum Hamiltonian density from the Lagrangian density in the usual manner we remember that the former has to be Hermitian so that: We can check the consistency of our definitions of the the fermionic momenta by looking at the Hamilton's equations for the fermionic variables: We thus see that is in agreement with our definitions of the Fermionic momenta. Also, for the , means an anti-commutator. We thus see that the anti-commulators in the above equation are non-trivial and need to be fixed. In order to fix these, we demand that c satisfy the Heisenberg equation: and using the property 2 The last three equations then imply : Here the minus sign in the above equation is nontrivial and implies the existence of states with negative norm in the space of state vectors of the theory.
The BRST charge operator Q is the generator of the BRST transformations. It is nilpotent and satisfies 2 0 Q = . It mixes operators which satisfy Bose and Fermi statistics. According to its conventional definition, its commutators with Bose operators and its anti-commutators with Fermi operators for the present theory satisfy: which at the instant-form time By imposing the conditions (obtained earlier): we then obtain Now let 0 denote the fermionic vacuum for which Defining 0 to have norm one, the last three equations imply † † The theory is thus seen to possess negative norm states in the fermionic sector.
The existence of these negative norm states as free states of the fermionic part of BRST  is however, irrelevant to the existence of physical states in the orthogonal subspace of the Hilbert space. In terms of annihilation and creation operators BRST  is: We also have and we further impose the dual condition that both Q and Q annihilate physical states, implying that: The states for which the constraints of the theory hold, satisfy both of these conditions and are in fact, the only states satisfying both of these conditions, since with Alternatively, one can understand the above point in terms of fermionic annihiliation and creation operators as follows. The condition 0 Q ψ = implies the that the set of states annihilated by Q contains not only the states for which the constraints of the theory hold but also additional states for which the constraints do not hold. However, 0 Q ψ = guarantees that the set of states annihilated by Q contains only the states for which the constraints hold, simply because † 0 G ψ ≠ and † 0 F ψ ≠ . Thus in this alternative way also, we see that the U. Kulshreshtha et al. states satisfying 0 Q Q ψ ψ = = are only those states which satisfy the constraints of the theory and we also see that these states belong to the set of BRST-invariant states as well as to the set of anti-BRST invariant states. This completes the BRST formulation of the theory.

Summary and Discussion
In the present work, we have considered the restricted gauge theory of quantum chromodynamics (QCD) in one-space one-time dimension (QCD 2 ) à la Cho et al. [1]- [14]. We have summarized the basics of the theory in Section 2 where the motivations of our present studies have also been discussed and are being omitted here the sake of brevity. The theory under our present investigation is seen to be GI and we have studied its quantization using Hamiltonian [24], path integral [25] [26] [27] [28] and Becchi-Rouet-Stora and Tyutin (BRST) [29] [30] [31], formulations [24]- [31], in the usual instant-form (IF) of dynamics (on the hyperplanes: 0 constant x t = = ) [32] [33], under appropriate gauge-fixing conditions.
The restricted gauge theory of QCD 2 à la Cho et al. [1]- [14] and others [1]- [23] makes use of the so-called "Cho-decomposition", which is, in fact, the gauge independent decomposition of the non-Abelian potential into the restricted potential and the valence potential and it helps in the clarification of the topological structure of the non-Abelian gauge theory. This decomposition allows one to view QCD as the restricted gauge theory (made of the restricted potential) which is coupled to a gauge-covariant colored vector field (the valence potential). The restricted potential is defined in such a way that it allows a covariantly constant unit isovector n everywhere in space-time, which enables one to define the gauge-independent color direction everywhere in space-time and at the same time allows one to define the magnetic potential of the non-Abelian monopoles. It even has full SU(2) gauge degrees of freedom, in spite of the fact that it is restricted. Consequently, the restricted QCD made of the restricted potential describes a very interesting dual dynamics of its own, and plays a crucial role in the understanding of QCD. This restricted gauge theory of QCD is therefore very important and it is important to study its quantization using the standard constraint quantization methods, including the Hamiltonian, path integral and BRST quantizations, as we have done in the present work.