Hamiltonian Formalism for Space-time Non-commutative Theories

Space-time non-commutative theories are non-local in time. We develop the Hamiltonian formalism for non-local field theories in d space-time dimensions by considering auxiliary d+1 dimensional field theories which are local with respect to the evolution time. The Hamiltonian path integral quantization is considered and the Feynman rules in the Lagrangian formalism are derived. The case of non-commutative \phi^3 theory is considered as an example.


I. INTRODUCTION
Space-time non-commutative field theories have peculiar properties due to their acausal behavior [1,2] and lack of unitarity [3]. In reference [4] it has been shown that there is a relation between lack of unitarity and the obstruction to finding a decoupling limit of string theory in an electromagnetic background [5][6][7][8][9][10]. These theories have an infinite number of temporal and spatial derivatives, and therefore are non-local in time and space [11,12]. The initial value problem of a non-local theory requires to give a trajectory or a finite piece of it [13]. The Euler-Lagrange (EL) equation is a constraint in the space of trajectories.
The Hamiltonian formalism for non-local theories was presented in [15]. In this paper we improve the formalism by clarifying the relation among the Lagrangian and Hamiltonian structures. We first consider an equivalent theory in a space-time of one dimension higher than that of the original theory. This space has "two times" and the dynamics is described in such a way that the evolution is local with respect to one of the times. For this equivalent theory one can construct the Hamiltonian. A characteristic feature of the Hamiltonian formalism for non-local theories is that it contains the EL equations as Hamiltonian constraints.
The Hamiltonian path integral for the d+1 dimensional field theory is constructed. The Lagrangian path integral formalism for the d dimensional theory is obtained by integrating out the momenta.
We apply the Hamiltonian formalism to time-like and light-like non-commutative theories [4]. As an example we consider the case of non-commutative φ 3 theory in d dimensions with space-time non-commutativity. The action contains the free Klein-Gordon Lagrangian and the interaction Lagrangian refers to the Moyal product. We construct the Hamiltonian in d + 1 dimensions. In the path integral quantization we get the Feynman rules that coincide with those used in references [1][2][3]. The theory is unitary at the classical level (tree level) but is not unitary at one loop [3]. This analysis should shed new light on the structure of these theories. The knowledge of the Hamiltonian of time-like and light-like non-commutative field theories could also be useful to study the energy of their solitons.

II. EULER-LAGRANGE EQUATIONS FOR NON-LOCAL THEORIES
Unlike standard Lagrangians, that depend on the values of a finite number of derivatives at a given timeq(t),q(t), . . . q (n) (t)-, a non-local Lagrangian depends on a whole piece of the trajectory q(t + λ), for all values of λ, that is, L non (t) = L([q(t + λ)]). At best it can be written as a function of all time derivatives q (j) (t) , j = 0, 1, 2, . . . at the same t. This means that the analog of the tangent bundle for Lagrangians depending on positions and velocities is infinite dimensional. The action is The EL equation is obtained as the variation of functional (1) and is given by where E(t, t ′ ; [q]) = δL non (t) δq(t ′ ) .
The EL equation must be understood as a functional relation to be fulfilled by physical trajectories. It is not a differential system, as one is used to find for local Lagrangians. In the latter case, the theorems of existence and uniqueness of solutions enable to interpret the EL equation as ruling the time evolution of the system, whose state at every instant of time is represented by a point in the space of initial data, e.g., J L = {q,q, . . . , q (2n−1) } for a local Lagrangian of order n.
In the non-local case, if we denote the space of all possible trajectories as J = {q(λ), λ ∈ R}, (2) is a Lagrangian constraint defining the subspace J R ⊂ J of physical trajectories.

III. 1 + 1 DIMENSIONAL FIELD THEORY DESCRIPTION OF NON-LOCAL THEORIES
Nevertheless, if we insist in defining a "time evolution" T t for a given initial trajectory q(λ), a natural choice is: We shall hence introduce new dynamical variables Q(t, λ) such that Thus, t is the "evolution" parameter and λ is a continuous parameter indexing the degrees of freedom. These new variables follow the evolution (4) above, and Q(0, λ) can be seen as initial data in the local 1 + 1 dimensional field theory. In differential form, condition (4) reads: where 'dot' and 'prime' respectively stand for ∂ t and ∂ λ . We then consider the Hamiltonian system for the 1 + 1 dimensional field Q with the Hamiltonian where P is the canonical momentum of Q. The phase space is thus T * J with the fundamental Poisson brackets In the Hamiltonian The "density" L(t, λ) is constructed from L non (t) by replacing q(t) by Q(t, λ), the t-derivatives of q(t) by λderivatives of Q(t, λ) and q(t + ρ) by Q(t, λ + ρ). In this construction of the Hamiltonian λ inherits the signature of the original time t and is a time-like coordinate.
Furthermore the symmetry of the original Lagrangian is realised canonically in the enlarged space [14]. Note that L(t, λ) is local in t and is non-local in λ. H depends linearly on P (t, λ) but does not depend onQ(t, λ). The relation (5) naturally arises as the first Hamilton equation for (6). However there is no a priori relationship between P (t, λ) and Q(t, λ) -unlike it happens in the local case-, the second Hamilton equation: does not imply any further restriction on Q(t, λ). Thus, the Hamiltonian system (6) on T * J is not so far equivalent to the non-local Lagrangian system of L non (t). Now, instead of taking the whole phase space T * J, we shall restrict to the subspace defined by the 1-parameter set of primary constraints [15]: where E(t; σ, λ) and χ(λ, −σ) are defined by Here ǫ(λ) is the sign distribution. The symbols "≡" and "≈" respectively stand for "strong" and "weak" equality. Further constraints are generated by requiring the stability of the primary ones. In the first step, we obtain: is the secondary constraint. Further constraints then follow by successive time differentiations of ψ 0 . They can be written all together in a condensed form as: Therefore, the constrained Hamiltonian system defined by the Hamiltonian (6) and the primary constraints (10) lives in a reduced phase space Γ ⊂ T * J defined by (10) and (14). Taking into account (4), the constraint (14) reduces to the EL equation (2) obtained from L non (t).
The constraints (10) and (14) belong to the second class in non-singular systems. In the next section we will show explicitly, for (non-singular) higher derivative Lagrangian system of order n, they are used to reduce the phase space to 2n dimensions reproducing the canonical Ostrogradski formalism [16]. Our formalism developed here turns out to be a generalization of the Ostrogradski formalism to the case of infinite order derivative theories. The infinite chain of second class constraints has also appeared in the description of boundary conditions as constraints [17]. Summarizing, the equivalence has been built in the 1 + 1 dimensional Hamiltonian formalism of local field theories through the constraints (10) and (14). This type of equivalence between the Hamiltonian and Lagrangian formalism is different from the one in local theories [18].

IV. NON-SINGULAR HIGHER ORDER DERIVATIVE THEORIES
Here we would like to derive both the Lagrangian and Hamiltonian formalisms for non-singular higher order derivative theories from the Hamiltonian formalism of non-local theories developed in the last section [19].
Let us consider a regular higher derivative theory described by the Lagrangian L(q,q,q, ..., q (n) ) and write the expressions obtained in the previous section for the nonlocal Lagrangian. As we embed the higher order theory in the non-local setting we start with the infinite dimensional phase space T * J(t) = {Q(t, λ), P (t, λ)}. They are assumed to be expanded in the Taylor basis [20] as where e ℓ (λ) and e ℓ (λ) are orthonormal basis The coefficients in (15) are new canonical variables and the Hamiltonian (6) is The momentum constraint (10) becomes an infinite set of constraints where On the other hand the constraint ψ in (14) in terms of the Taylor basis becomes These constraints (19) and (21) are second class and are used to reduce the infinite dimensional phase space to finite one leading to the ordinary Ostrogradski Hamiltonian formalism. The operator D t defined in (20) becomes a time evolution operator for q's using the first set of Hamilton equationq Using this in (19) the constraints ϕ m , (0 ≤ m ≤ n − 1) coincide with the definition of the Ostragradsky momenta , Now they can be solved for q ℓ , (n ≤ ℓ ≤ 2n − 1) as functions of canonical paires {q j , p j }, (0 ≤ j ≤ n − 1) They are combined with the constraints ϕ ℓ , (n ≤ ℓ ≤ 2n − 1) to form a second class set and can be used to eliminate the canonical pairs {q ℓ , p ℓ } (n ≤ ℓ ≤ 2n − 1). If we take into account (22) the constraint (21) for m = 0 is the Euler-Lagrange equation for the original higher derivative Lagrangian, The constraints (21) for m > 0 are the time derivatives of the Euler-Lagrange equation (26) expressed in terms of q's. For a non-singular theory, all the constraints (21) can be rewritten as q ℓ − q ℓ (q 0 , q 1 , ..., q n−1 , p 0 , p 1 , ..., p n−1 ) ≈ 0, (ℓ ≥ 2n) (27) and can be paired with the constraints ϕ ℓ , (ℓ ≥ 2n) forming second class constraints. They are used to eliminate canonical paires {q ℓ , p ℓ } (ℓ ≥ 2n). In this way the infinite dimensional phase space is reduced to a finite dimensional one. The reduced phase space is coordinated by T * J n = {q l , p l } with l = 0, 1, ..., n − 1. All the constraints are second class and we use the iterative property of Dirac bracket. The Dirac bracket for these variables have the standard form, The Hamiltonian (6) in the reduced space is given by where q n is expressed using (24) as a function of the reduced variables in T * J n . Note that if we consider the limit n going to infinity the constraints (19) and (21) do not allow, in general, to reduced the dimensionality of the infinite dimensional phase space of the non-local system via Dirac brackets.

V. SYMPLECTIC FORMULATION OF THE EULER-LAGRANGE EQUATION
The Hamiltonian formalism presented in the last sections can be cast into a symplectic form as follows. The Poisson brackets (7) correspond to the symplectic two form Ω ∈ Λ 2 (T * J): where δ stands for the functional exterior derivative.
In the constrained phase space Γ 1 ⊂ T * J defined by (10) only, the induced (pre)symplectic form is: The induced Hamiltonian is: The generator of the time evolution (4) is the vector field Now, i(X)Ω 1 + δH 1 = 0 gives a first order formulation of the EL equation. Indeed a short calculation yields i(X)Ω 1 + δH 1 = − dσE(t; σ, 0) δQ(t, 0) Whence, the evolution (5) and the EL equation (2) follow from it.

VI. PATH INTEGRAL QUANTIZATION
Let us consider the Hamiltonian path integral quantization of the 1 + 1 dimensional field theory associated with the Hamiltonian (6) for L non (t). The path integral is given by The integration is performed over the reduced phase space Γ and the measure [21,22] µ is First we consider the non-singular higher derivative Lagrangian system of order n. From the discussions of section IV the constraints are arranged to a set in which the canonical variables (q j , p j ) for j ≥ n are expressed in terms of ones for 0 ≤ j < n. The measure becomes where ... terms are given functions of (q i , p i ), (i = 0, ..., n − 1). Integrating over (q i , p i ), (i ≥ n) (37) becomes where q n is given as a function of (q i , p i ), (i = 0, ..., n−1). This is the Hamiltonian path integral of the Ostrogradski formalism. If we assume non-local systems can be regarded as infinite n limit of higher derivative system (40) becomes, by taking n → ∞, where Q and P , which are n → ∞ of (q i , p i ), (i = 0, ..., n − 1), are not restricted by the constraints in contrast to (37). If we integrate out the momenta and using δ(Q(t, λ) − Q ′ (t, λ)) we get which is the Lagrangian path integral formulation for the non-local theory.

VII. APPLICATION TO SPACE-TIME NON-COMMUTATIVE φ 3 THEORY
Space-time non-commutative theories have peculiar properties due to their acausal behavior and lack of unitarity. Here we would like to use the previous formalism to study the question of unitarity in these theories.
To fix the ideas we consider a non-commutative φ 3 theory with arbitrary non-commutativity in d dimensions. The Lagrangian density is given by where * is the star product defined by using a general anti-symmetric background θ µν , The EL equation is x 0 in (43)-(45) will be denoted as t hereafter. We introduce a "new coordinate x 0 " which plays the role of λ in the previous discussion and introduce the field Q(t, x µ ) in d + 1 dimensions. Now t is regard as "evolution time" and x µ := (x 0 , x) is a continuous Lorentzian index. Our metric conventions are η tt = η 00 = −1, η ii = +1. The relation (4) in this case is The Lagrangian density in d + 1 dimensions for Q(t, x µ ), (see eq.(8)), is where now the derivatives in * are with respect to x µ = (x 0 , x). Note that this Lagrangian density is local in the evolution time t. The momentum constraint (10) is given by where Q ′ (t, x) denotes ∂ x 0 Q(t, x µ ). K is the symmetric kernel of three star products, The Hamiltonian (6) is The Hamilton equations arė The stability of the constraint implies the new constraints (13) By requiring further consistency we have an infinite number of constraints which can be written as (14) ψ(t, x) = 0 for − ∞ < x 0 < ∞.
Using Hamilton equation (51) for Q, (54) becomes the EL equation where ∂ x 0 on Q is replaced by ∂ t both in the first term and in the * product. It is the original non-local EL equation (45). If we write the symplectic form and the Hamiltonian in terms of Q(t, x), eqs. (34) and (32), we have and H = dx δ(x 0 ) 2 {Q ′ (t, x) 2 + (∇Q(t, x)) 2 + m 2 Q(t, x) 2 } − g 4 dx(Q(t, x) * Q(t, x))ǫ(x 0 )Q ′ (t, x).
These expressions can be rewritten in terms of φ(x) using (51) i.e. (46). In particular the interaction Hamiltonian becomes Note that the occurrence of time derivatives of any order in the interaction Hamiltonian is not forbidden in nonlocal theories. This property is clearly not fulfilled by local theories. Now we can perform the path integral quantization using (42) to obtain From which we read the Lagrangian Feynman rules, [23]. They coincide with the ones used in [3]. Therefore, it follows from [3] and [4] that non-commutative φ 3 theory with time-like non-commutativity is not unitary while non-commutative φ 3 theory with light-like noncommutativity is unitary.
Note Added: Recently, reference [24] has also considered the Hamiltonian formalism for non-local theories.