Virasoro Algebra for Particles with Higher Derivative Interactions

In this paper we show that the worldline reparametrization for particles with higher derivative interactions appears as a higher dimensional symmetry, which is generated by the truncated Virasoro algebra. We also argue that for generic nonlocal particle theories the fields on the worldline may be promoted to those living on a two dimensional worldsheet, and the reparametrization symmetry becomes locally the same as the conformal symmetry.


Introduction
Due to technical difficulties, physicists have been reluctant to consider higher derivative interactions. Yet they are unavoidable in many important physical problems. For a partial list, see [1]. In string theory, for example, we do not fully understand how to deal with higher derivative terms in the worldsheet action as they are nonrenormalizable, even though such background interactions should exist. String field theory is another outstanding example of nonlocal theory. For some recent interests in nonlocal theories in the context of string theory, see [2].
In fact, even for particles, higher derivative interactions are not well understood. In this paper we study a very basic property of particle worldline theory -the reparametrization symmetry, for worldline Largrangians with higher derivatives. We find that, remarkably, while the phase space of higher derivative theories are of higher dimensions, the reparametrization group acting on the phase space is also of higher dimension. Furthermore, for generic nonlocal theories, the reparametrization group is locally equivalent to the conformal group of two dimensions. It will be very interesting to see whether a new class of well-defined conformal field theories can originate from nonlocal worldline theories.
The plan of this paper is as follows. We first review the worldline theory of a charged particle and define its higher derivative generalization (sec.2). Then we show that both the phase space and the reparametrization group are of higher dimensions for higher derivative theories (sec.3). In the nonlocal limit, the reparametrization group becomes the conformal group (sec.4). Finally we comment on the connection to the string theory and the generalization of general covariance, as well as on the issue of stability (sec.5).

Worldline Lagrangian
Let us start by reviewing the worldline theory of a charged particle moving in curved spacetime with metric g µν where e = e(τ ) is the tetrad of the worldline and φ is the potential term. For particles of mass m without external potential, φ = m 2 . The action is invariant under the reparametrization of the worldline parameter τ It induces a transformation of the worldline fields e and x δe = d dτ (ǫe), and δx = ǫẋ.
The symmetry of general coordinate transformation in the gravitational theory is reflected in the particle theory as the invariance of the action under a field redefinition x µ → x ′µ (x) and a simultaneous transformation of the coefficients g µν and A µ . The U(1) symmetry of the gauge potential A µ , on the other hand, appears as changing the Lagrangian by a total derivative.
The only generator of the reparametrization symmetry is the Hamiltonian H. In the usual covariant quantization scheme, we fix the gauge by setting e = 1 and impose the constraint H = 0.
Let us now consider the worldline Lagrangian in its full generality. The most general particle Lagrangian L(x,ẋ,ẍ, · · · ; e) is an arbitrary function of x µ and all of their time derivatives. We can expand L by the number of time derivatives as where is the covariant derivative of τ . For a given order n of D and a set of integers P (n) = {P k ≥ 0; k = 1, · · · , n} satisfying n k=1 kP k = n, there is a spacetime field A (P 1 ,···,Pn) (x) coupled to the particle by the interaction invariant under (3), where m = n k=1 P k .

Reparametrization Symmetry
We start with Lagrangians (4) involving only finite order time derivatives. Assume that the Lagrangian (4) is a function of {x (n) ; n = 0, 1, ·, N}, where we used the notation f (n) ≡ d dτ n f (τ ), and that it is nondegenerate. To apply canonical quantization, we can introduce new variables x 1 , · · · , x N −1 and add to the Lagrangian, where λ i 's are the Lagrange multipliers. This trick allows us to replace x (n) by x n for n = 1, 2, · · · , N − 1 in the Lagrangian, which then becomes a function of x 0 , x 1 , · · · , x N −1 andẋ N −1 , with only first time derivatives. Dirac's constrained quantization can be applied straightforwardly.
The equations of motion of the x's are differential equations of order 2N, which require 2N initial data to determine a solution. One can think of the phase space as the space of initial conditions given at τ = 0. Hence the phase space for each x is 2N dimensional.
The fact that we have a larger phase space for higher derivative theories has significant implications. Consider the worldline reparametrization symmetry (3). Taking its k-th derivative and evaluating it at τ = 0, we get Although these expressions follow directly from (3), they are independent transformations on the phase space for k = 0, 1, 2, · · · , N − 1, since x,ẋ, etc. are independent variables. Eq. (8) defines N independent transformations with parameters ǫ(0),ǫ(0), · · · , ǫ (N −1) (0). Naively, even for k ≥ N, eq.(8) still looks like a transformation on the phase space. But it is inconsistent to treat them as independent transformations because they do not preserve the Poisson structure. In other words, it is impossible to find operators to generate these transformations.
The above can be derived more rigorously as follows. We add (7) to the Lagrangian and replace x (n) 's by x n 's. The Lagrangian now has only first derivatives of x. In order for L ′ to be invariant under the reparametrization, we need The 2nd transformation law in (9) agrees with (8) and can be conveniently summarized as where we only need the first N terms in the Taylor expansion Note that λ n is identified with p n−1 , the conjugate momentum of x (n−1) , in the constrained quantization. So (9) also tells us how the momenta transforms.
The N symmetry generators corresponding to ǫ (k) are We see that the reparametrization symmetry is N dimensional for a particle theory with N-th time derivatives. A redefinition of the H k 's (12) to Note that this is a consistent truncation of the classical Virasoro algebra. This is one of the main results of this paper. Roughly speaking, in the N → ∞ limit, we expect to obtain only "half" of the Virasoro algebra. In order to have the full algebra, one has to consider nonlocal theories which are not defined as the large N limit of finite derivative theories.

Conformal Symmetry for Nonlocal Particle
In this section we consider generic nonlocal particle theories. As various kinds of nonlocal theories may differ significantly in nature, in order for a generic discussion without specifying details of the theory, this section will be more heuristic and less rigorous than before. We do not expect everything in this section to hold for all nonlocal theories, but we believe that our results will apply to a wide class of examples. In fact, it will be very interesting to have any nonlocal particle theory for which the reparametrization symmetry coincides with the conformal symmetry.
To deal with Lagrangians with infinite derivatives, an interesting proposal [3,4,5] is to introduce an auxiliary coordinate σ as follows. For x(τ ) we introduce X(τ, σ) and impose the constraintẊ where X ′ ≡ d dσ X. This guarantees that X can be identified with the original variable as X(τ, σ) = x(τ + σ). Since the generic action (4) also involves higher derivatives of the tetrad e(τ ), it should also be promoted to E(τ, σ), satisfying a constraint like (14). The covariant derivative D (5) will be replaced by D σ = E −1 (τ, σ)∂ σ . when acting on X. By replacing all D by D σ and x(τ ) by X(τ, σ) in L 0 , we obtain a new Lagrangian L 1 which has the same equation of motion if the constraint (14) is satisfied.
The constraint (14) can be imposed by adding to L 1 . This can also be obtained as an N → ∞ limit of (7). The new Lagrangian L ≡ L 1 + L ′ does not have any higher time derivatives, but only higher derivatives of the auxiliary coordinate σ. The worldline reparametrization (2) of the original Lagrangian L 0 induces the reparametrization of σ δσ = −ǫ(τ + σ), leaving τ invariant. The derivative D σ and the scalar field X should be invariant under (16), that is, f (τ, σ) = (f + δf )(τ, σ + δσ) for f = D σ and X. It follows that the transformation of E(τ, σ) and X(τ, σ) are given by To have L 1 invariant under (16), we also need the measure of integration E(τ, σ)dτ ∧dσ to be invariant, which can be easily verified.
In addition, we need to impose certain boundary conditions at the boundary values of σ (denoted σ 0 , σ 1 ) such that The boundary condition can be chosen to be This is also an appropriate condition to guarantee that the equation of motion for the new Lagrangian L is equivalent to the original equation of motion. By varying X and λ in the new Lagrangian L, we get (14) anḋ where (EOM) is the expression for the equation of motion for the original Lagrangian L 0 , but with x replaced by X, etc. Using (14) we can replace X(τ, σ) by x(τ + σ), but (22) does not guarantee that (EOM) = 0. Instead, because (EOM) is a function of (τ + σ) only, (22) implies that for an arbitrary function f . Applying the boundary condition (21) to it, we find two identities valid for all τ . They imply λ = 0 and (EOM) = 0 as we hoped.
As the reparametrization symmetry (16) is labelled by a one-variable function ǫ at τ = 0, it is locally the same as the conformal symmetry in two dimensions.
So far we have restricted ourselves to the classical theory. In principle the quantum anomaly (the central charge of the Virasoro algebra) can appear in the large N limit, but we need further details of the theory in order to treat it with some rigor.

Remarks
Connection to String Theory: Naively, higher derivative particle Lagrangians can be related to the boundary string field theory [6] as the open string worldsheet Lagrangian in the zero metric limit. If this interpretation is acceptable, the boundary string field theory can be interpreted as the theory over "the space of all particle theories" instead of "the space of all open string boundary theories". One can also view strings as a convenient technique to summarize the infinite degrees of freedom of particles with nonlocal interactions. However, it is hard to make this story rigorous due to the technical problem that higher derivative terms in the string worldsheet action are nonrenormalizable.
Another possible connection with string theory is to generalize the Seiberg-Witten (SW) limit [7] for open strings ending on D-branes in a constant B field background. Since the open string vertex operator for the U(1) field interaction involves only the first time derivative, only the zero mode of the string survives in the SW limit. If other open string vertex operators with higher derivatives are considered, more degrees of freedom of the string will survive the analogous SW limit, if it exists.

Symmetry of Spacetime Fields
In the infinite derivative limit, we can generalize the story of general covariance and U(1) symmetry mentioned in the paragraph below eq.(3) in sec.2. The particle Lagrangian (4) is invariant under a simultaneous worldline field redefinition and a complicated, mixed transformation of the fields A (P ) . This defines a generalized notion of general covariance. Similar tranformations for the string coordinates has been used to construct string field theory [8], and it would be interesting to see if they have any connection with the higher spin gauge theories [9]. Generalization of the U(1) symmetry is straightforward. The Lagrangian is always defined only up to a total derivative d dτ Λ(x,ẋ,ẍ, · · ·), which induces a gauge transformation of the A P (n) 's.

Stability Problem
Higher derivative theories are known to have various problems such as stability and causality. Since the notion of causality may be changed at the Planck scale where spacetime is fuzzy, we will only comment on the problem of stability. Theories with higher derivatives of finite order suffer the Ostrogradskian instability because the canonical Hamiltonian is always unbounded from below [10]. For low energy effective theories, the perturbative formulation [1] might be a convenient way to avoid these problems. On the other hand, stable nonlocal theories are known to exist. At least those theories obtained from integrating out certain physical fields of a stable field theory should still be stable.
In this paper we studied worldline theories with higher derivatives. Although the Hamiltonian is constrained to vanish, and thus is not unbounded from below, an attempt to couple it to other physical systems may still lead to instability. Ideally, this work should have been focused on nonlocal theories without instability. We hope this paper may serve as part of the motivation for the task of understanding how to construct sensible nonlocal theories.

Note Added in Revision
After the first version of this paper appeared, two earlier works [11,12] were brought to my attention. These works considered nonlocal particle Lagrangians closely related to the explicit example of this paper. The work in Ref. [11] did not mention reparametrization symmetry, and their model is not exactly the same as the string. In the other work [12], a quantization different from the canonical quantization of the original variables was chosen such that it is equivalent to the open string. It was also argued there that the reparametrization symmetry is equivalent to the conformal symmetry for quadratic Lagrangians. In this paper, we have provided a comparatively more general and more explicit discussion on the reparemetrization symmetry.
I was also informed that, in a series of papers [13], generic particle Lagrangians with only first derivatives are considered as the starting point of formulating the "generalized equivalence principle". Our proposal for "generalized general covariance" described above is simply a generalization of that to include higher derivatives. This is a crucial difference when comparing the corresponding higher spin gauge theory with string theory. Only totally symmetrized tensor fields appear in the first deriviative theory of [13], but we know that in string theory there are many more tensor fields. Our model contains all the tensor field degrees of freedom in open string theory.