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The Path Integral Quantization And The Construction Of The S-matrix In The Abelian And Non-Abelian Chern-Simons Theories

The cvariant path integral quantization of the theory of the scalar and spinor particles interacting through the abelian and non-Abelian Chern-Simons gauge fields is carried out and is shown to be mathematically ill defined due to the absence of the transverse components of these gauge fields. This is remedied by the introduction of the Maxwell or the Maxwell-type (in the non-Abelian case)term which makes the theory superrenormalizable and guarantees its gauge-invariant regularization and renormalization. The generating functionals are constructed and shown to be formally the same as those of QED (or QCD) in 2+1 dimensions with the substitution of the Chern-Simons propagator for the photon (gluon) propagator. By constructing the propagator in the general case, the existence of two limits; pure Chern-Simons and QED (QCD) after renormalization is demonstrated. By carrying out carefully the path integral quantization of the non-Abelian Chern-Simons theories using the De Witt-Fadeev-Popov and the Batalin-Fradkin- Vilkovisky methods it is demonstrated that there is no need to quantize the dimensionless charge of the theory. The main reason is that the action in the exponent of the path integral is BRST-invariant which acquires a zero winding number and guarantees the BRST renormalizability of the model. The S-matrix operator is constructed, and starting from this S-matrix operator novel topological unitarity identities are derived that demand the vanishing of the gauge-invariant sum of the imaginary parts of the Feynman diagrams with a given number of intermediate on-shell topological photon lines in each order of perturbation theory. These identities are illustrated by an explicit example.


Introduction
The past fifteen years witnessed an increasing interest in the theories of matter coupled Chern-Simons (CS)gauge field theories in 2+1 dimensions. From one point of view, the Euclidean version of such theories can be viewed as giving the high temperature bahaviour of 3+1 dimensional models [1]. On the other hand, in the pioneering works [2,3] it has been shown that the introduction of the ( P and T odd ) CS term into the Lagrangian of 2+1 dimensional QED and QCD, leads to a very peculiar property : the gauge field splits into two parts; a massive part (that acquires a mass in a gauge-invariant manner), and a massless part which does not contribute to the free classical Hamiltonian, but leads to an additional interaction among the particles. This interaction appears also in pure CS theories [4].
In the work [3], it was argued that in the non-Abelian version of CS theories, the dimensionless combination of the charge and the stochastic parameter should be quantized. It was also shown that the mass term provides an infrared cut-off in special covariant gauges that renders the theory superrenormalizable.
Many works were devoted to the consideration of the one-loop radiative corrections to the charge and the stochastic parameter in both the Abelian [5] and the non-Abelian [6] theories, and a theorem [7] was set which states that under very general conditions, there are no further radiative corrections beyond the finite one-loop for these parameters.
An additional thrust into the interest in CS theories was provided by the interesting results in the non-relativistic domain; essentially the idea of Wilczeck that non-relativistic charged particles coupled to pure CS field can be considered as a phenomenological approach for the description of the "bound states " of two particles called anyons [8]. This idea found wide acceptance, and many attempts to apply it in many interesting condensed matter phenomena, such as the fractional quantum Hall effect, and high temperature superconductivity were made ( see the reviews [9] and the references therein ). CS theories also found applications in the field-theoretic formulation of the Aharonov-Bohm effect [10,11].
One of the issues that received considerable interest during the past period was the canon-ical quantization of the CS models [2,3,12]. However, some interesting points like the canonical quantization in a Lorentz covariant gauge still need further investigation. Path integral quantization was also considered first -up to our knowledge-in the works [11,13] where the generating functional was also constructed.
Another issue that did not receive much attention is the following: The free transverse topological photons of the pure CS theory are absent, while the gauge field propagator is present, and gives significant contribution to the interaction among the particles. This issue was addressed in the work [14], and the so called topological unitarity identities were derived. Moreover, the issue of the quantization of the charge in non-abelian CS theories was not discussed thoroughly beyond the discussion in the works [2,3]. We address this point in the present work.
This paper is a further development of the series of works [11,13,14]. The main goals are, to carry out the path integral quantization and construct the generating functional for a wide class of models involving both the Abelian and the non-Abelian CS fields (part II), to construct the S-matrix operator, and to develop the Feynman rules and formulate a Wick-type theorems for the CS field (part III), and to illustrate in details the topological unitarity identities in general, and through a specific example ( part IV). Part V is devoted to concluding remarks.

Path Integral Quantization and the Generating Functional
The aim of this part is to develop the path integral quantization, and to construct the generating functional of the theory of scalar and spinor fields interacting through the Abelian and non-Abelian CS field in 2+1 dimensions. This can be done through two different approaches : The De Witt-Fadeev-Popov (DFP) [15] approach, or the Batalin-Fradkin-Vilkovisky (BFV) approach [16]. The latter was developed to quantize gauge theories with both classes of constraints and with arbitrary constraint algebra. In our case both approaches lead to the same result . This is a consequence of the fact that the first class constraints, both in the Abelian and non-Abelian cases, form a closed algebra, and that the structure functions in the algebra of the first class constraints are just constants, as will be demonstrated later. Therefore, we shall carry out the path integral quantization through the simpler DFP approach, and will prove the equivalence of both approaches by invoking the latter in the quantization of the theory of spinors interacting through the non-abelian CS gauge field. This proof is very helpful in understanding the connection between the usual canonical quantization and the BFV quantization schemes, and in the demonstration of the appearence of the BRST operators of the theory.

De Witt-Fadeev-Popov method
Scalar particles: We begin with the theory of charged scalar particles interacting through the CS gauge field with the action of this gauge field given slolely by the abelian CS term (pure CS field). Following the DFP method, we get for the generating functional in the covariant α-gauge the expression [11,13] : where Here, J µ (x), j(x) and j * (x) are external sources, e and m are respectively the charge and the mass of the scalar field, and D µ = (∂ µ − ieA µ ). The metric is taken as g µν = diag(1, −1, −1) . The Greens functions of the theory are defined as usual by varying the above generating functional, eq.(1) with respect to the sources. For example, the free propagator of the CS field is defined as: However, the functional (1) has an essential defect in that the path integral over the gauge field is not mathematically well-defined . This is because in pure CS theory, there are no transverse components of the gauge field. The natural way to overcome this difficulty is to introduce into the total action in the exponent of the path integral (1) the Maxwell term Such a term is the only gauge-invariant bilinear term in A µ that guarantees gauge-invariant regularization and renormalization of the theory. This term, not only leads to the convergence of the path integral over A µ , but also plays the role of a regularization factor since the resulting theory becomes superrenormalizable [2,3].
It is necessary here to make some important remarks on the dimensions of the parameters and the fields of the theory. We have some arbitrariness in the choice of the dimensions of the statistical parameter µ, the charge e and the factor γ in eqs.(3),(5) and (7). However, if we require the 2+1 dimensional matter-coupled CS theory to have some relation with the real world, and so that it arises after compactification on the ∼ 1 γ layer of QED in 3+1 dimensions [17]with the parity violating term µ 4 F µνF µν d 4 x wherẽ F µν = 1 2 ε µνλσ F λσ then the charge e and the parameter µ are to be chosen dimensionless, and [γ] = x −1 . In the following, we will adopt this convention of the dimensions 1 . So, after introducing the Maxwell term the generating functional takes the form This can be formally written in the alternative form 1 If one makes the change of variables Aµ → A ′ µ = Aµ √ γ , e → e ′ = e √ γ , µ → µ ′ = µ γ then one gets the conventions used in the works [2,3] whereS m does not contain the term e 2 A µ A µ in eq.(5),i.ẽ After integrating over A µ in eq.(9) we get : where and D µν (x − y) is the CS gauge field's Greens function defined by the equation: or, The solution of eq.(14) is [2,3]: We note that the above Greens function (or propagator) consists of two parts: The first two terms describe the propagation of a real massive photon with mass equal to γµ; the third term describes the propagation of a topological massless photon, and the last term is pure gauge term. The appearence of massive photons in a gauge-invariant manner is a well-known peculiar property of CS theory, and is independent of coupling to matter fields [2,3]. To show that the topological term in eq. (15) does not contribute to the tensor F µν of the gauge field, we construct the general solution of the classical equations of motion of the field A µ (eq. (14)). This is given as : Here, ImD µν (p) is the imaginary part of the propagator D µν in eq.(15) in the momentum space representation; e ν δ (p), δ = 0, 1, 2, are three mutually orthogonal polarization vectors which satisfy p µ e µ δ (p) = 0. This choice corresponds to the gauge ∂ µ A µ = 0. In the general case, the free solution A µ (x) in eq.(16) represents the sum of two independent parts : The terms proportional to δ(p 2 − µ 2 γ 2 ) correspond to a real massive photon which contributes to the free Hamiltonian; the fourth and fifth terms are the topological parts of the gauge field which do not contribute to the classical free Hamiltonian, but give nontrivial contribution to the propagator (see eq.(15)) , and the last term is merely a gauge term that can be removed by a gauge transformation. It is easy to see that the topological part of A µ does not contribute to F µν : multiplying both sides by ε σµν we get As for the the massive part of the solution (16), excluding the second term in this equation in view of (19) above, then we have for the massive part and this gives a non-vanishing contribution to F µν . We shall return later to the question of quantization of this A µ in connection with the construction of the S-matrix of the theory (see part III).
Returning to the general expression for the Greens function of the gauge field , we stress that formally it is possible to consider two limiting procedures in eq. (15). First, if γ → ∞ we obtain: which is just the propagator of the pure CS theory. In the limit µ → 0, we get the usual Feynman propagator in 2+1 dimensional QED for massless photons: In both cases, we have from eq.(16) A µ as The limits are to be taken after renormalization, recalling that in our theory one has only the finite one-loop correction to µ (or γ) [5][6][7].
Spinor CS theory: Let us consider now the theory of spinor particles interacting through the CS gauge field. The DFP method gives the following expression for the generating functional in this case [14]: where Z 0 = Z(0, 0, 0) ; S CS , S g and S M are defined by eqs. (3),(4) and (7) respectively, and where, and the Dirac matrices are defined as where σ's are the Pauli spin matrices. The γ-matrices satisfy ψ(x) andψ(x) = ψ(x) † γ 0 are the two-component Grassmann spinors, η andη are Grassmann sources. Integrating over A µ (x) in eq.(25) we get : and D µν (x − y) is the bare CS field propagator which is the same as in the scalar case, eq. (15). Here also, as in the scalar case, one can consider the limits (after renormalization) γ → ∞ and µ → 0 to get the propagators of pure CS field and 2+1 dimensional QED respectively.
Non-Abelian CS theory: The path integral quantization of theories with the non-Abelian CS gauge field is a bit more complicated than the Abelian one, so we consider it in some detail. We start with the theory of the gauge field without coupling to matter , i.e CS gluodynamics, defined by the Lagrangian L M is the usual Yang-Mills Lagrangian in 2+1 dimensions, L CS is the non-Abelian CS term The gauge group is SU (N ). In matrix notation The t a 's are antihermitian matrices in the fundamental representation of the group f abc are the structure constants of the SU (N ) group.
To see the difference of the non-Abelian case from the Abelian one, consider a general gauge transformation If we suppose that at ||x|| = x 2 0 + x 2 → ∞ , A µ → 0 faster than 1 ||x|| then the second term in (38) vanishes. The last term , however, coincides in euclidean space, with the so called homotopy class or winding number, and is equal to 0, ±1, ±2, ... . This result follows from the fact that if then 3-dimensional space can be mapped onto S 3 ; for SU (2) group U (x) realizes the mapping S 3 → S 3 and the winding number is equal to the degree of mapping S 3 to the SU (2) group. On the classical level, the gauge-transformation (37) results in It is clear that this constant does not influence the equations of motion or any physical quantity.
Now, we use the Fadeev-Popov trick to quantize the theory. Formally, the vacuum functional of the theory is where N is a normalization factor that will be defined later. Introducing into the formal equation (42) the identity operator in a general Lorentz covariant gauge where Dµ(G) is the measure of the SU (N ) group, and Eq.(43) defines the Fadeev-Popov determinant △(A).
We know that in perturbation theory we can forget about the Gribov ambiguity [18] and consider only contributions to the functional integral from elements near the identity of the group G; where λ a (x) is infinitesimally small for all x. This means that in DFP method we must consider gauge transformations which belongs to the zero homotopy class for which w = 0 since λ(x) must go to zero when ||x|| → ∞ 2 . Substituting the identity operator (43) into the expression (42), we get after the conventional manipulations Here Ω(G) is the infinite group volume, and where C(x) andC(x) are the well-known Fadeev-Popov ghosts that are scalar Grassmann fields, and Thus, the generating functional of the theory is now given by the expression: here, With the above generating functional, the expectation value of any observable is well- This expression does not change under any gauge transformation of the total action .
We have seen above that the exponent in the generating functional of the theory 2 In the general case when U = e iτ a λ a (x) and ||x|| → ∞ , λ(x) = (λ a ) 2 → 2πn where n is the winding number.
contains after quantization the term S g that violates gauge-invariance. However, the total action in the exponent preserves invariance under a special class of BRST [19] gauge where ǫ is an x-independent Grassmann parameter (ǫ 2 = 0); It is well-known that S g does not change under these transformations. If we formally write down the transformation law of A µ (x) in the form then under this transformation since w = 0 because ǫ 2 = 0.
It is very important to stress that the BRST-invariance of the CS theory ensures satisfying all the Ward-Fradkin-Takahishi-Slavnov-Taylor identities [20], and therefore the gaugeinvariant renormalizability of the theory [21]. We thus come to the conclusion that in the framework of perturbation theory, it is not necessary to quantize the charge in CS gluodynamics. The same holds also if coupling to matter is introduced into the theory as well.
If one introduces spinor field into the theory (CS quantum chromodynamics (CSQCD)), then it is straight forward to generalize the generating functional eq.(49) to this case. The resulting expression is Here S CS , S M and S g were defined earlier,eqs. (3), (4) and (7),and i, j = 1, ..., N above are the color indices of the SU (N ) group in the fundamental representation.It is straight forward to write down the Feynman propagator of the non-Abelian gauge field; it will differ from the abelian one only by the appearence of color indices viz.
Before leaving this subsection, we would like to emphasize that starting from the generating functionals for the various models that have been considered so far, one can construct all the propagators and the primitive vertices, and thus develop the Feynman rules for perturbation theory. For example, the Feynman propagator for the scalar field is given as Similarly, we have for the spinor propagator from eq.(59)

Path Integral Quantization Of Pure Chern-Simons Quantum Chromodynamic by Batalin-Fradkin-Vilkovisky Method
Here, we shall show how to construct the generating functional of the theory of spinors coupled to the non-Abelian CS field by the BFV method. We shall consider however, a theory where the gauge field kinetic action is given solely by the non-abelian CS term, i.e pure CSQCD. This will have a more complicated constraint structure than the one with the Maxwell term (eq. (7)) included.The BFV quantization makes the BRST symmetry of the theory, that is generated by the operator Ω introduced below, more transparent. We start with the classical action: where S CS and S ψ are given by eqs. (3) and (26). The action can be written in a more transparent form: The canonical momenta of the theory turn out to be all primary constraints: The standard Poisson brackets are: θ i , θ 3 and θ 4 are second class constraints, while G a is first class. The presence of the second class constraints motivates one to define the Dirac brackets [22] using these constraints.
These can be worked out easily, and the ones that differ from the Poisson bracket are: The Hamiltonian assumes the form H 0 is the Hamiltonian on the constraint surface, and T a is a first class constraint that is the analogue of Gauss' law constraint in QCD , and here also it is the generator of the gauge symmetry. A 0 appears here, as is the case in QED and QCD, as a Lagrange multiplier. The first class constraint T a can be seen to satisfy the algebra : The BFV quantization method, in attempting to maintain Lorentz covariance and the unitarity of the S-matrix expands the phase space of the theory by making the Lagrange multiplier of the theory dynamical, and introducing new (ghost) degrees of freedom whose statistics are oppositte to the first class constraints of the theory. In our case we will have two pairs of these ghosts which are Grassmann fields ; (C a ,P a ) ; (P a ,C a ).
Therefore, our canonical variables become now Generally, the BFV method introduces the so called complete Hamiltonian [16] that enters into the expresion of the generating functional, which is defined as where χ a is a gauge-fixing function The vacuum functional of the theory is given now by the expression where P A and Q A are given in eqs.(79) and (80), and Ber is the superdeterminant, or the Berezinian, which is introduced here due to the presence of the fermionic degrees of freedom. Integrating over the matter and gauge momenta and over π a 0 ,P b and P a we get and,C The above expression -upon including external sources-coincides with the generating functional eq.(59) without the Maxwell term .

The S-Matrix Operator
Although the generating functionals of the theory, eqs.
coincides with the classical equation (14) and have the same solution, eq.(15). However, in the case of pure CS theory, this topological photon does not contribute to the physical states of the Hilbert space, which can be defined as usual; ∂ µÂ + µ |phys = 0. Thus, starting from this result, one can unambigously formulate rules for the construction of any matrix elements of the different products of this operator. In this sense, one can formulate some kind of Wick theorem for the operators of the topological CS photon. The S-matrix operator for scalar pure CSQED has been construsted in the works [11,13], and those for spinor CSQED in [14]. Here, we would like to elaborate on the construction given in these references.
In pure CSQED, the S-matrix operator formally has the same form as that in 2+1 dimen- here, where ": :" means normal ordering, andψ(x) andψ(x) operators are given aŝ ) are respectively the annihilation and creation operators of particles (antiparticles) satisfying the usual anticommutation relations The two-component spinors u(p), v(p) are respectively the positive and negative energy solutions of the free Dirac equation in (2+1) dimensions, with the properties: Let us next our attention to the operatorÂ µ : Using its above mentioned properties, we can formulate the following rules of the matrix elements of its products : 1) The vacuum expectation value of the products and the T-products of only an even number of the operators A µ is nonvanishing, and reduces respectively to the sum of the vacuum expectation values of the product and the T-product of two field operators defined as: where D µν (x − y) is given by eq. (21). For example, for four operator product we have: and so on. 2) All the matrix elements between physical states of the normal product of any number of the field operators A µ are equal to zero . However, the vacuum expectation value of the product of the normal products of equal number of these operators only is different from zero. For example: and so on.
Thus, the above rules are the same as the Wick rules except that we take into account the absence of physical states with free topological photons (other than the vacuum state !). Therefore, we make now the following observation: All the Feynman rules of the theory are identical to those of QED given that one replaces the Maxwell propagator in internal lines by the CS propagator, and excludes diagrams with external photon lines. In mathematical language, the above rules mean that the total set of physical states in the total Hilbert space of the theory does not contain states with real free topological photons 3 , but only the physical states of particles and antiparticles. The interesting consequences and applications of these statements will be considered in part IV.
Consider now the more general case of MCSQED, where the propagator is given by eq. (15) and the free field solutions of the classical equations of motion by eq. (16). This solution consists of two parts : massive physical part and massless topological part. The canonical quantization of the massive part in the α = 0 gauge can be carried out, and gives the following representation for the physical massive partÂ m The S-matrix in this case looks formally the same as (93), but the Wick theorem is now the usual one 0| :Â µ (x)Â ν (y) :: and so on, where D µν (x− y) is given by eq. (15).Only one important exception exists : Any matrix element of the normal product of the operators A µ reduces to that of the normal product of the massive operators A m µ ; Here, |i and |j are two arbitrary physical states of the total Hilbert space of the theory. Now the total set of physical states includes, in addition to spinor particles, real massive photons, but never the topological massless photons.
The generalization of the S-matrix operator to scalar or spinor pure CSQCD is straight forward now. For the spinor case, the generating functional is given by eq.(90). The S-matrix will have the form The Wick-type theorem for the operatorsψ,ψ,Ĉ,Ĉ is as usual. As for theÂ a µ operator, we have the same rules as in the Abelian case, except that the Greens function will have now an additional kronecker delta in the color indices.

Topological Unitarity Identities
In this part we are going to investigate the consequences of the peculiar property of the CS theories, namely the absence of real topological photons in spite of the presence of the propagator and the many-particle Greens function of the gauge field that contribute to the interaction of the particles quantum mechanically ( It is well-known that on the classical level, the CS field do not contribute to the interaction of the particles !). We will see that the above property of the CS theories leads upon imposing the unitarity condition on the theory to very interesting topological unitarity identities. These identities have been derived in the work [14]. Here, we essentially follow the development in this reference, however, we discuss in more details how do these identities hold in the general case when the Maxwell term is present along with the CS term.
We consider first the case of CSQED. The propagator is given by eq. (21) , and the Smatrix operator is given by eq.(92). As we have mentioned above, the absence of the real CS photons means that the complete set of physical vector states in the total Hilbert space of the theory does not contain these topological particles. To investigate the consequences of this fact, we introduce theT -matrix : whereŜ is the S-matrix operator (the energy-momentum conserving δ-function has been suppressed). The unitarity of the S-matrix operator leads to the well-known relation: For arbitrary non-diagonal (|i = |f ) on-shell matrix elements between two physical states of the total Hilbert space, we can write the two equivalent relations and, ) .
where G µν (k) = ε µνλ k λ , and δ + (k 2 ) = θ(k 0 )δ(k 2 ). For simplicity, we restrict ourselves to the case of forward scattering in which case the imaginary part of these diagrams give their contribution to the total cross-section of the process. As was shown in [14],a lengthy calculation gives (an overall irrelevant multiplicative constant has been suppressed) or, The same result can be obtained in the case of non-forward scattering too. This example demonstrates the unitarity identities in the one-loop order.
It is not difficult to generalize the unitarity identities to the case when Maxwell-type terms are present. In such cases, one must divide the total gauge field propagator in eq.(15) or The identities developed above can be also shown to hold outside the framework of perturbation theory. That they should hold in the non-Abelian case as well, could be demonstrated without too much difficulty.

Concluding Remarks
In this paper, we have shown that the covariant path integral quantization of the theories of scalar and spinor particles interacting through the Abelian and non-Abelian pure CS gauge fields, is mathematically ill-defined due to the absence of the transverse components of these gauge fields. To define the path integral, it is necessary to introduce into the classical action the Maxwell or Maxwell-type (in the non-Abelian theory) term that is the only bilinear term in the gauge field that does not violate the gauge-invariance of the action. This term also guarantees the gauge-invariant regularization and renormalization of the theory, which becomes then superrenormalizable [2,3].
The generating functionals of the various models considered were constructed, and seen to be formally the same as those of QED (or QCD) in 2+1 dimensions, with the substitution of the CS gauge field propagator for the photon (or gluon) propagator. The CS propagator in these models is seen to consist of two parts: the first part is the propagator of a real massive photon (gluon) which contributes to the classical free Hamiltonian, and its states appear in the Hilbert space of the total set of physical states of the system.
The second part is that of the topological massless photon which does not contribute to the free Hamiltonian, but leads to additional (in comparison with QED or QCD) interaction between the charged particles. The general solution for the free gauge field, when constructed in a covariant gauge, was therefore seen to consist of a massive part, and a massless topological part.
Taking the limits γ → ∞ and µ → 0 of the propagators and the general solutions of the gauge fields (see eqs.(21)-(24) ) after renormalization, which is possible due to the finite renormalization of these parameters [5][6][7], we get respectively pure CSQED and QED in 2+1 dimensions.
We carried out very carefully the path integral quantization of some models with the Unfortunately, a path integral representation of the S-matrix is not available for theories with the pure CS field. This is because the "in" and "out" limits of the transverse part of the pure CS gauge field do not exist. In the general case, when the Maxwell-type term is included in the action, such a representation can be constructed, and this will depend only on the "in" and "out" solutions of the massive part of the gauge field. We constructed in the general case, the S-matrix operator for all the Abelian and non-Abelian models, and showed that this operator gives the correct expression for all the Feynman diagrams of the theory, and formally differs from the usual case of QED and QCD in 2+1 dimensions only by a specific type of Wick theorem for the gauge field.
Starting from this S-matrix operator, we have shown that the requirement of the unitarity of the S-matrix leads to topolgical unitarity identities that were derived in [14]. These