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Multimomentum Hamiltonian Formalism in Quantum Field Theory

The familiar generating functionals in quantum field theory fail to be true measures and, so they make the sense only in the framework of the perturbation theory. In our approach, generating functionals are defined strictly as the Fourier transforms of Gaussian measures in nuclear spaces of multimomentum canonical variables when field momenta correspond to derivatives of fields with respect to all world coordinates, not only to time.


Introduction
Contemporary field models are almost always the constraint ones. In order to describe them, one can apply the covariant multimomentum generalization of the Hamiltonian formalism in mechanics [8,9,11]. The multimomentum canonical variables are field functions φ i and momenta p λ i associated with derivatives of φ i with respect to all world coordinates x µ , not only the time.
In classical field theory, if a Lagrangian density is degenerate, the system of the Euler-Lagrange equations becomes underdetermined and requires additional conditions. In gauge theory, these are gauge conditions which single out a representative from each gauge class. In general case, the above-mentioned supplementary conditions remains elusive. In the framework of the multimomentum Hamiltonian formalism, one obtaines them automatically because a part of the Hamilton equations play the role of gauge conditions. The key point consists in the fact that, given a degenerate Lagrangian density, one must consider a family of associated multimomentum Hamiltonian forms in order to exaust solutions of the Euler-Lagrange equations [8,9]. There exist the comprehensive relations between Lagrangian and multimomentum Hamiltonian formalisms for degenerate quadratic and affine densities. The most of field models are of these types. As a result, we get the general procedure of describing constraint systems in classical field theory.
The present work is devoted to the multimomentum quantum field theory. This theory as like as the well-known current algebra models, has been hampered by the lack of satisfactory commutation relations between multimomentum canonical variables [2,4]. We are based on the fact that the operation of chronological product of quantum bosonic fields is commutative and, so Euclidean chronological forms can be represented by states on commutative tensor alebras. Therefore, restricting our consideration to generating functionals of Green functions, we can overcome the difficulties of establishing the multimomentum commutation relations.
Moreover, the multimomentum quantum field theory may incorporate together the canonical and algebraic approaches to quantization of fields. In physical models, the familiar expression of a generating functional fails to be a true measure since the Lebesgue measure in infinite-dimensional linear spaces is not defined in general.
In the algebraic quantum field theory, generating functionals of chronological forms result from the Wick rotation of the Fourier transforms of Gaussian measures in the duals to nuclear spaces [1,6,7]. The problem has consisted in constructing such measures. In the present work, we get these measures in terms of multimomentum canonical variables. They have the universal form due to the canonical splitting of multimomentum Hamiltonian forms. In particular, we reproduce the Euclidean propagators of scalar fields and gauge potentials [10]. Note that the covariant multimomentum canonical quantization can be generalized to any field model with a degenerate quadratic Lagrangian density.

Multimomentum Hamiltonian Formalism
We consider the multimomentum generalization of the familiar Hamiltonian formalism to fibred manifolds π: E → X over a n-dimensional base X, not only X = R. If sections of E describe classical fields, one can apply this formalism to field theory. In this case, the Legendre plays the role of the finite-dimensional phase space of fields. By V E and V * E are meant the vertical tangent and cotangent bundles over a fibred manifold E. Given an atlas of fibred coordinates (x µ , y i ) of E, the Legendre manifold is provided with the linear adapted coordinates (x µ , y i , p µ i ). In this coordinates, a multimomentum Hamiltonian form on Π and the corresponding Hamilton equations read where Γ is a connection on E and Hω is a horizontal density on Π → X. The multimomentum Hamiltonian formalism is associated with the Lagrangian formalism in jet manifolds where the jet manifold J 1 E of E plays the role of a finite-dimensional configuration space. The jet manifold J 1 E comprises classes j 1 x φ of sections φ of E which are identified by the first two terms of their Taylor series at points x. It is provided with the adapted coordinates (x λ , y i , y i λ ) where A first-order Lagrangian density L = L(x λ , y i , y i λ )ω on J 1 E defines the Legendre morphism L of J 1 E to Π: Conversely, a multimomentum Hamiltonian form H on Π defines the momentum morphism H of Π to J 1 E: We say that a multimomentum Hamiltonian form H is associated with a Lagrangian density L if In general, different multimomentum Hamiltonian forms may be associated with the same Lagrangian density. The most of field models meet the following relations between Lagrangian and multimomentum Hamiltonian formalisms.
(i) All multimomentum Hamiltonian forms H associated with a Lagrangian density L are equal to each other on the constraint space Q, that is, H| Q = H L . Moreover, for every section φ of E, we have (ii) If a solution r of the Hamilton equations (3) corresponding to a multimomentum Hamiltonian form H associated with a Lagrangian density L belongs to the constraint space Q, then π Π • r is a solution of the Euler-Lagrange equations for L. Conversely, for each local solution s of the Euler-Lagrange equations defined by a Lagrangian density L, there exists an associated multimomentum Hamiltonian form such that L • s is a solution of the corresponding Hamilton equations.
The relation (4) gives the reason to use sections r of the Legendre manifold Π → X as functional variables in quantum field theory. On physical level, one can consider the naive generating functional Note that the canonical splitting (2) of multimomentum Hamiltonian forms leads to standard terms p µ i ∂ µ φ i in generating functionals in multimomentum canonical variables.
The generating functional (5) fails to be a true measure. The problem of representation of generating functionals by measures can be settled in the framework of the algebraic quantum field theory [6,7].

Algebraic Quantum Field Theory
In accordance with the algebraic approach, a quantum field system can be characterized by a topological * -algebra A and by a continuous state f on A. To describe particles, one considers usually a tensor algebra A Φ of a real linear locally convex topological space Φ endowed with involution operation. We further assume that Φ is a nuclear space.
In the axiomatic quantum field theory of real scalar fields, the quantum field algebra is is finite for any collection (α 1 , . . . , α m ) and all l, k ∈ Z ≥0 . A state f on A RS 4 is represented by a family of temperate distributions W n ∈ S ′ (R 4n ): If f obeys the Wightman axioms, W n are the familiar n-point Wightman functions.
To describe particles created at some moment and destructed at another moment, one uses the chronological forms f c given by the expressions where (i 1 . . . i n ) is a rearrangement of numbers 1, . . . , n. The forms (6) fail to be distributions and do not define a state on A RS 4 . At the same time, they issue from the Wick rotation of the Euclidean states on A RS 4 describing particles in interaction zone.
Since chronological forms (6) are symmetric, Euclidean forms can be introduced as states on a commutative tensor algebra. Note that they differ ¿from the Swinger functions associated with the Wightman functions.
Let B Φ be the commutative quotient of A Φ . This algebra can be regarded as the enveloping algebra of the Lie algebra associated with the Lie commutative group G Φ of translations in Φ. We therefore can construct a state on the algebra B Φ as a vector form of its cyclic representation induced by a strong-continuous unitary cyclic representation of G Φ . Such representation is characterized by a positive-type continuous generating function Z on Φ, that is, for all collections of φ 1 , . . . , φ n and complex numbers α 1 , . . . , α n . If the function α → Z(αφ) on R is analytic at 0 for each φ ∈ Φ, the positive continuous form F on B Φ is given by the expressions In virtue of the well-known theorem [3], any function Z of the abovementioned type is the Fourier transform of a positive bounded measure µ in the dual Φ ′ to Φ: where , denotes the contraction between Φ ′ and Φ. The corresponding representation of G Φ is given by operators in the space of quadraticly µ-integrable functions u(w) on Φ ′ , and we have F n (φ 1 · · · φ n ) = w, φ 1 · · · w, φ n dµ(w).
For instance, a generating function Z of a Gaussian state F on B Φ reads where the covariance form Λ(φ 1 , φ 2 ) is a positive-definite Hermitian bilinear form on Φ, continuous in φ 1 and φ 2 . This generating function is the Fourier transform of a Gaussian measure in Φ ′ . The forms F n>2 obey the Wick rules where In particular, if Φ = RS n , the covariance form of a Gaussian state is uniquelly defined by a distribution W ∈ S ′ (R 2n ): In field models, a generating function Z plays the role of a generating functional represented by the functional integral (7). If Z is the Gaussian generating function (8), its covariance form Λ defines Euclidean propagators. Propagators of fields on the Mincowski space are reconstructed by the Wick rotation of Λ [11].

Scalar Fields
Let E be a vector bundle over a world manifold X 4 . Its sections describe scalar matter fields. In jet terms, their Lagrangian density reads where a E is a fibre metric in E, Γ is a linear connection on E and g is a world metric on X 4 . Because of the canonical vertical splitting V E = E × E, the corresponding Legendre manifold (1) is The Legendre morphism L (m) and the unique multimomentum Hamiltonian form associated with the Lagrangian density L (m) are given by the expressions where a E is the fibre metric in E * dual to a E . For the sake of simplicity, we here examine scalar fields without symmetries Let φ be real Euclidean scalar fields on the Euclidean space X = R 4 . The corresponding Legendre manifold is provided with the adapted coordinates (z µ , y, p µ ). Sections r of Π are represented by functions ( φ(z), p µ (z)) on R 4 which take their values in the vector space Their commutative tensor algebra is brings Φ into the rigged Hilbert space. Let be the multimomentum Hamiltonian form describing Euclidean scalar fields. The covariance form Λ of the associated generating function is defined by the relation where γ is a first order linear differential operator on Φ. We have where ∆ F is the Feynman propagator of Euclidean scalar fields.

Gauge Theory
Let P → X 4 be a principal bundle with a structure Lie group G of internal symmetries. There is the 1:1 correspondence between principal connections on P and global sections A C of the affine bundle C = J 1 P/G modelled on the vector bundle The bundle C is provided with the fibred coordinates (x µ , k m µ ) such that its section A C has the coordinate expression where A m µ (x) are coefficients of a local connection 1-form. In gauge theory, sections A C are treated as gauge potentials.
The configuration space of gauge potentials is the jet manifold J 1 C. It is provided with the adapted coordinates (x µ , k m µ , k m µλ ). There exists the canonical splitting where c m nl are the structure constants of the Lie algebra g of the group G. In the coordinates (13), the conventional Yang-Mills Lagrangian density L (A) of gauge potentials is given by the expression where a G is a G-invariant metric in the Lie algebra g and g is a world metric on X 4 . For gauge potentials, we have the Legendre manifold provided with the canonical coordinates (x µ , k m µ , p µλ m ). This is a phase space of gauge potentials. It also has the canonical splitting The fibred manifold Π → X is the affine bundle modelled on the vector bundle The Legendre morphism corresponding to the Lagrangian density (14) is The multimomentum Hamiltonian forms associated with the Lagrangian density (14) read where B is some section of C, Γ is a connection on C and Γ β µλ are Christoffel symbols of a world metric g. We have The Hamilton equations corresponding to the multimomentum Hamiltonian form (17) read plus the equation (16b). On the constraint space (16a), the equations (16b) and (18) are the familiar Yang-Mills equations. The equation (19) plays the role of gauge condition. In the algebraic quantum field theory, only fields forming a linear space are quantized. We therefore fix a background gauge potential B and consider deviation fields Ω = A C − B which are sections of the vector bundle (12). The corresponding Legendre bundle (15) is endowed with the adapted coordinates Sections r of the Legendre bundle (15) over the Euclidean space R 4 are represented by functions (Ω m µ (z), p µλ m (z)) taking their values in the vector space The commutative tensor algebra of r is B Φ where Φ = F ⊗ RS 4 . The nuclear space Φ is provided with the corresponding scalar form | Φ which brings Φ into the rigged Hilbert space.
To define a Gaussian state on this algebra, let us consider the multimomentum Hamiltonian form (17). In the coordinates (20), it reads where F B is the strength of the background gauge potential B, and Γ B is a connection on C associated with the principal connection B.
One can use the multimomentum Hamiltonian form (21) in order to quantize the deviation fields Ω on X = R 4 . For the sake of simplicity, let us assume that the structure group G is compact and simple (a mn G = −2δ mn ). We have where ∇ denotes the covariant derivative corresponding to the principal connection B.
To construct the associated Gaussian state, we use the quadratic part r * H 1 of the form (22). The term r * H 2 describes interaction considered by the pertubation theory.
The scalar form r * H 1 on Φ however is degenerate. There are two ways for this difficulty to be overcomed.
(i) In accordance with the conventional quantization scheme, we can restrict ourselves to sections r taking values in the constraint space (16a). Since the form is degenerate, we must then consider the gauge orbit space Ξ G which is the quotient of the space Ξ of connections A C by the group of gauge transformations. There exists a neighborhood N centered at the image of B in Ξ G such that there is a local section s B : N → Ξ whose values are elements A C ∈ Ξ satisfying the gauge condition [5]. Hence, N is locally isomorphic to a Hilbert space. Not discussing here the Gribov ambiguity problem, let us assume that there exists a connection B such that s B is a global section. If the bilinear part r * H 0 of the form (23) induces a nuclear scalar form on Ξ ′ G , one can construct the associated Gaussian measure µ in Ξ G . If s B is µ-measurable morphism, there exists a measure µ GF in Ξ which is the image of µ with respect to s B . If this measure exists, it is concentrated in s B (Ξ G ) ⊂ Ξ. We call it the gauge-fixing measure. In contrast with the naive expressions used in the gauge models, it is not the measure whose base is a Gaussian measure and density is the Faddeev-Popov determinant. Determinant densities are attributes of Lebesgue measures which fail to be defined in general case.
(ii) The first procedure fails to be applied to general case of degenerate field systems, without gauge invariance. At the same time, one can insert additional terms quadratic in p (µν) m into the multimomentum Hamiltonian form (21) which bring r * H 1 into a nondegenerate scalar form. In general case, we have [a 1 δ µν δ λβ δ mn p (µλ) m p (νβ) n + a 2 (p µµ n ) 2 ], where a 1 = 0 and a 2 are some constants. The Lagrangian L ′ (A) associated with the multimomentum Hamiltonian form (24) includes additional terms quadratic in k The associated covariance form Λ B exists. In particular, it defines the propagator of Euclidean deviation fields Ω which coinciders with the Green's operator of the covariant Laplasian δ µν ∇ µ ∇ ν . This Green's operator exists [5]. In the case of B = 0, the propagator of deviation fields Ω coinciders with the familiar propagator of gauge potentials which corresponds to the Feynman gauge (α = 1), but there are no ghost fields. In comparison with the measure µ GF , the Gaussian measure µ B defined by the generating function Z B is not concentrated in a gauge-fixing subset. Note that, after gauge transformations B → B ′ , measures µ B and µ B ′ are not equivalent in general. It means that a gauge phase of a background gauge potential may be valid, otherwise electromagnetic potentials. In the case of an abelian structure group G, the multimomentum Hamiltonian form (23), the associated covariance form Λ B and the Gaussian measure µ B are independent on a background potential B.