Eisenhart lift and Randers-Finsler formulation for scalar field theory

TL;DR

This paper extends Eisenhart lifts and Randers-Finsler formulations from point particle mechanics to scalar field theories and Nambu-Goto actions, providing new geometric insights into these theories.

Contribution

It introduces a novel extension of Eisenhart lifts and Randers-Finsler formulations to scalar field and string actions, broadening their applicability.

Findings

Extended Eisenhart lift to scalar field theory.

Developed Randers-Finsler formulation for Nambu-Goto action.

Provided a Jacobi Lagrangian for string actions.

Abstract

We study scalar field theory as a generalization of point particle mechanics using the Polyakov action, and demonstrate how to extend Lorentzian and Riemannian Eisenhart lifts to the theory in a similar manner. Then we explore extension of the Randers-Finsler formulation and its principles to the Nambu-Goto action, and describe a Jacobi Lagrangian for it.

Equations

$$S={\int }_{\mathcal{V}}{d}^{n}x\sqrt{-g}\mathcal{L}(g_{\mu \nu },{\phi }^i,{\partial }_{\mu }{\phi }^i),{\partial }_{\mu }{\phi }^i=\frac{\partial {\phi }^i}{\partial x^{\mu }}$$

$$\begin{array}{rl}\delta S& ={\int }_{\mathcal{V}}{d}^{n}x\sqrt{-g}[\left(\frac{1}{2}{g}^{\mu \nu }\mathcal{L}+\frac{\partial \mathcal{L}}{\partial g_{\mu \nu }}\right)\delta g_{\mu \nu }\\ & +\left\{\frac{\partial \mathcal{L}}{\partial {\phi }^i}-{\nabla }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\phi }^i)}\right)\right\}\delta {\phi }^i+{\nabla }_{\mu }\left(\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\phi }^i)}\delta {\phi }^i\right)].\end{array}$$

$$\frac{\partial \mathcal{L}}{\partial {\phi }^i}-{\nabla }_{\mu }\left({\mathcal{P}}_i^{\mu }\right)=0,\text{where }{\mathcal{P}}_i^{\mu }:=\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\phi }^i)}.$$

$$\mathcal{L}=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})h_{ij}(\mathbf{\phi }){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j.$$

$$g^{\mu \nu }(\mathbf{x})\left({\nabla }_{\mu }{\partial }_{\nu }{\phi }^i+{\Omega }_{jk}^i{\partial }_{\mu }{\phi }^j{\partial }_{\nu }{\phi }^k\right)=0,$$

$${\Omega }_{jk}^i=\frac{h^{im}(\mathbf{\phi })}{2}\left(\frac{\partial h_{mj}(\mathbf{\phi })}{\partial {\phi }^k}+\frac{\partial h_{mk}(\mathbf{\phi })}{\partial {\phi }^j}-\frac{\partial h_{jk}(\mathbf{\phi })}{\partial {\phi }^m}\right).$$

$${\nabla }_{\mu }{\mathcal{P}}_{\mathcal{V}}^{\mu }={\partial }_{\mu }{\mathcal{P}}_{\mathcal{V}}^{\mu }+{\Gamma }_{\mu \rho }^{\mu }{\mathcal{P}}_{\mathcal{V}}^{\rho }=\frac{\partial \mathcal{L}}{\partial \mathcal{V}}=0.$$

$$\mathcal{L}=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})\left[h_{ij}(\mathbf{\phi }){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j+2h_{i0}(\mathbf{\phi }){\partial }_{\mu }{\phi }^i{\partial }_{\nu }\mathcal{V}+h_{00}(\mathbf{\phi }){\partial }_{\mu }\mathcal{V}{\partial }_{\nu }\mathcal{V}\right].$$

$$\begin{array}{rl}{\mathcal{P}}_{\mathcal{V}}^{\mu }=\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }\mathcal{V})}& =g^{\mu \nu }(\mathbf{x}){\left[h_{i0}(\mathbf{\phi }){\partial }_{\nu }{\phi }^i+h_{00}(\mathbf{\phi }){\partial }_{\nu }\mathcal{V}\right]}_{\mathcal{V}(\mathbf{x})=\mathcal{V}(y)}\\ & =g^{\mu \nu }(\mathbf{x})h_{i0}(\mathbf{\phi }){\partial }_{\nu }{\phi }^i+g^{\mu 0}(\mathbf{x})h_{00}(\mathbf{\phi }){\partial }_0\mathcal{V}.\end{array}$$

$$g^{\mu 0}(\mathbf{x})=h_{i0}(\mathbf{\phi })=0,\mathcal{V}=\mathcal{V}(y)\Rightarrow {\mathcal{P}}_{\mathcal{V}}^{\mu }=0\forall \mu \ne 0.$$

$${\partial }_0{g}_{\mu \nu }=0\Rightarrow {\Gamma }_{\mu 0}^{\mu }=\frac{1}{2}{g}^{\rho \sigma }{\partial }_0{g}_{\rho \sigma }=0.$$

$${\nabla }_{\mu }{\mathcal{P}}_{\mathcal{V}}^{\mu }={\nabla }_0{\mathcal{P}}_{\mathcal{V}}^0={\partial }_0{\mathcal{P}}_{\mathcal{V}}^0=\frac{\partial \mathcal{L}}{\partial \mathcal{V}}=0\Rightarrow {\mathcal{P}}_{\mathcal{V}}^0=const.$$

$$G({\phi }^i,{\mathcal{P}}_i^{\mu }):={\mathcal{P}}_i^{\mu }{\partial }_{\mu }{\phi }^i-\mathcal{L}.$$

$$\begin{array}{rl}{\partial }_{\alpha }G& =\frac{\partial G}{\partial {\phi }^i}{\partial }_{\alpha }{\phi }^i+\frac{\partial G}{\partial {\mathcal{P}}_i^{\mu }}{\partial }_{\alpha }{\mathcal{P}}_i^{\mu }\\ & ={\partial }_{\mu }{\phi }^i{\partial }_{\alpha }{\mathcal{P}}_i^{\mu }+{\mathcal{P}}_i^{\mu }{\partial }_{\alpha }{\partial }_{\mu }{\phi }^i-\left(\frac{\partial \mathcal{L}}{\partial {\phi }^i}{\partial }_{\alpha }{\phi }^i+\frac{\partial \mathcal{L}}{\partial ({\partial }_{\mu }{\phi }^i)}{\partial }_{\alpha }{\partial }_{\mu }{\phi }^i\right).\end{array}$$

$$\frac{\partial G}{\partial {\phi }^i}{\partial }_{\alpha }{\phi }^i+\frac{\partial G}{\partial {\mathcal{P}}_i^{\mu }}{\partial }_{\alpha }{\mathcal{P}}_i^{\mu }={\partial }_{\mu }{\phi }^i{\partial }_{\alpha }{\mathcal{P}}_i^{\mu }-{\nabla }_{\mu }{\mathcal{P}}_i^{\mu }{\partial }_{\alpha }{\phi }^i,$$

$${\partial }_{\mu }{\phi }^i=\frac{\partial G}{\partial {\mathcal{P}}_i^{\mu }},{\nabla }_{\mu }{\mathcal{P}}_i^{\mu }=-\frac{\partial G}{\partial {\phi }^i}.$$

$$\mathcal{L}=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})h_{ij}(\mathbf{\phi }){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j+\frac{1}{2}{g}^{00}(\mathbf{x})h_{00}(\mathbf{\phi }){\left({\partial }_0\mathcal{V}\right)}^2,$$

$$G({\phi }^i,{\mathcal{P}}_i^{\mu },{\mathcal{P}}_{\mathcal{V}}^0)=\frac{1}{2}{g}_{\mu \nu }(\mathbf{x})h^{ij}(\mathbf{\phi }){\mathcal{P}}_i^{\mu }{\mathcal{P}}_j^{\nu }+\frac{1}{2g^{00}(\mathbf{x})h_{00}(\mathbf{\phi })}({\mathcal{P}}_{\mathcal{V}}^0{)}^2.$$

$${\mathcal{P}}_{\mathcal{V}}^0=g^{00}(\mathbf{x})h_{00}(\mathbf{\phi }){\partial }_0\mathcal{V}=q.$$

$$\Phi (\mathbf{\phi }):=\frac{q^2}{2g^{00}(\mathbf{x})h_{00}(\mathbf{\phi })},$$

$${\partial }_{\mu }{\phi }^i=\frac{\partial G}{\partial {\mathcal{P}}_i^{\mu }}=g_{\mu \nu }(\mathbf{x})h^{ij}(\mathbf{\phi }){\mathcal{P}}_j^{\nu },$$

$$\mathcal{L}=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})h_{ij}(\mathbf{\phi }){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j-\Phi (\mathbf{\phi }),$$

$$\mathcal{L}=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})h_{ij}(\mathbf{\phi }){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j+\frac{q^2}{4\Phi (\mathbf{\phi })}{\left({\partial }_0\mathcal{V}\right)}^2.$$

$$\mathcal{L}=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})h_{ij}(\mathbf{\phi },\mathcal{U}){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j+g^{00}(\mathbf{x})\left(\frac{1}{2}{h}_{\mathcal{UU}}(\mathbf{\phi },\mathcal{U})({\partial }_0\mathcal{U}{)}^2+h_{\mathcal{U}\mathcal{V}}(\mathbf{\phi },\mathcal{U})({\partial }_0\mathcal{U})({\partial }_0\mathcal{V})\right),$$

$$G({\phi }^i,\mathcal{U},{\mathcal{P}}_i^{\mu },{\mathcal{P}}_{\mathcal{U}}^0)=\frac{1}{2}{g}_{\mu \nu }(\mathbf{x})h^{ij}(\mathbf{\phi },\mathcal{U}){\mathcal{P}}_i^{\mu }{\mathcal{P}}_j^{\nu }+{\mathcal{P}}_{\mathcal{U}}^0{\mathcal{P}}_{\mathcal{V}}^0-\frac{1}{2}{h}_{\mathcal{UU}}(\mathbf{\phi },\mathcal{U})({\mathcal{P}}_{\mathcal{V}}^0{)}^2.$$

$$G({\phi }^i,\mathcal{U},{\mathcal{P}}_i^{\mu },{\mathcal{P}}_{\mathcal{U}}^0)=\frac{1}{2}{g}_{\mu \nu }(\mathbf{x})h^{ij}(\mathbf{\phi },\mathcal{U}){\mathcal{P}}_i^{\mu }{\mathcal{P}}_j^{\nu }+q{\mathcal{P}}_{\mathcal{U}}^0-\frac{q^2}{2}{h}_{\mathcal{UU}}(\mathbf{\phi },\mathcal{U}).$$

$$\begin{array}{rl}{\partial }_{\mu }{\phi }^i& =\frac{\partial G}{\partial {\mathcal{P}}_i^{\mu }}=g_{\mu \nu }(\mathbf{x})h^{ij}(\mathbf{\phi },\mathcal{U}){\mathcal{P}}_j^{\nu },\\ {\partial }_0\mathcal{U}& =\frac{\partial G}{\partial {\mathcal{P}}_{\mathcal{U}}^0}=q\Rightarrow \mathcal{U}(y)=qy.\end{array}$$

$$L={\mathcal{P}}_i^{\mu }{\partial }_{\mu }{\phi }^i+{\mathcal{P}}_{\mathcal{U}}^0{\partial }_0\mathcal{U}-G({\phi }^i,\mathcal{U},{\mathcal{P}}_i^{\mu },{\mathcal{P}}_{\mathcal{U}}^0)=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})h_{ij}(\mathbf{\phi },\mathcal{U}){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j-V(\mathbf{\phi },\mathcal{U}).$$

$$\mathcal{L}=\frac{1}{2}{g}^{\mu \nu }(\mathbf{x})h_{ij}(\mathbf{\phi },\mathcal{U}){\partial }_{\mu }{\phi }^i{\partial }_{\nu }{\phi }^j-\frac{1}{q^2}V(\mathbf{\phi },\mathcal{U})({\partial }_0\mathcal{U}{)}^2+({\partial }_0\mathcal{U})({\partial }_0\mathcal{V}).$$

$$S$$

Full text

1 Introduction

The application of geometric methods in physics led to modern theories at the heart of active research, such as String Theory and Braneworld Cosmology. The Kaluza-Klein theory is one such topic, where one adds extra dimensions to account for other interactions, while the Eisenhart lift is another that formulates a new background curvature that replaces the gauge fields that influence a particle’s motion by adding degrees of freedom associated with symmetry. The original idea discussed by Eisenhart in 1928 [1] was forgotten with minimal impact until independent rediscovery by Duval in 1985, and further developed and studied by Gibbons, Horvathy and others [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

The lifting concept is not completely new, being closely related to Jacobi-Maupertuis metric formulation [13, 14, 15], that does the opposite by formulating gauge fields from the curvature to replace the degree of freedom associated with symmetry. When a theory is potential free, the corresponding equation is tautologically geodesic. Adding potentials to the theory causes a deviation from geodesic. However, by using the Eisenhart lift, this can be embedded into the geodesic equation of the metric in higher-dimensional space with one extra fictious field.

Scalar field theory in n-dimensional field space is a generalization of point particle mechanics, which leads us to ask if we can also similarly perform an Eisenhart lift of the scalar field space. The main obstacle to performing the Eisenhart lift on field space is the divergence term involved in the field equations. Due to this, Finn, Karamitsos and Pilaftsis have demonstrated how to execute the Riemannian lift [16] of a scalar field space with Lagrangian of the Polyakov form by introducing vector fields.

The Nambu-Goto action [17, 18] is a geometric way to write the action for a string traversing spacetime, similar to the first term in the Randers-Finsler geodesic. While other extended objects can be considered for world volume actions, the string alone exhibits Weyl invariance. Since it is a geometric action, one can ask if and how the formulation and principles associated with Randers-Finsler geodesics discussed in [13] can be extended to it.

The goal of this paper is to study the Eisenhart lift of the scalar field space, and extend mechanical principles associated with point particle, world volume actions. We shall thoroughly describe both, the Riemannian and Lorentzian lifting processes, in a simple and original way to generalize the Eisenhart lift for application on scalar field space. Then the general geometric action for a world volume will be considered to study how the Maupertuis principle, constraint, a generator analogous to the Hamiltonian, and possibly a Jacobi-Maupertuis formulation may be described for it, with the specific case of a surface swept by a string being briefly discussed.

2 Preliminaries: Scalar Field Theory

While scalar field theory is a generalization of point particle mechanics, special circumstances in the latter allow a conserved quantity to exist due to an available cyclic co-ordinate, which is not normally possible in the former. In this section, we shall review scalar field theory formulation, discuss how a conserved quantity can exist if a cyclic field is available, and introduce a generator of field equations analogous to the Hamiltonian.

If we have a $N+1$-dimensional scalar field space $\{\varphi^{i}(\bm{x})\}$ and define the Lagrangian $\mathcal{L}=\mathcal{L}(g_{\mu\nu},\varphi^{i},\partial_{\mu}\varphi^{i})$ on the space, the field indices in configuration space $0\leq i,j,k,m\leq N$, co-ordinate indices in base space $0\leq\mu,\nu\leq n$ and the field action:

[TABLE]

then remembering that $\delta(\partial_{\mu}\varphi^{i})=\partial_{\mu}(\delta\varphi^{i})=\nabla_{\mu}(\delta\varphi^{i})$, the arbitrary variation of the action (2.1) in the spacetime region $\mathcal{V}$ can be expanded using chain rule:

[TABLE]

Thus, the field equation of motion for classical trajectories is given by:

[TABLE]

To see how classical mechanics for single point particle is a field theory as well, setting $x^{\mu}\rightarrow\tau\;\Rightarrow\;d^{n}x\rightarrow d\tau$, and $g_{\mu\nu}(\bm{x})\rightarrow g\rightarrow 1$ gives us the familiar action integral and the Euler-Lagrange equation for point particles.

When considering dynamics of $N+1$-dimensional scalar field space, we will essentially deal with the Polyakov form Lagrangian:

[TABLE]

and its Euler-Lagrange field equation given by (2.3) as:

[TABLE]

where the field space connection symbol $\Omega^{i}_{jk}$ in (2.5) is

[TABLE]

Now we shall proceed to discuss how a conserved quantity may exist in scalar field theory as a result of a cyclic field.

2.1 Cyclic fields

Even if the field Lagrangian $\mathcal{L}$ is independent of a cyclic field $\varphi^{0}=\mathcal{V}$, the corresponding field momentum $\mathcal{P}^{\mu}_{\mathcal{V}}$ is not necessarily a conserved quantity, since according to (2.3), we have a covariant divergence equation.

[TABLE]

Now consider the following field Lagrangian based on the model (2.4) where $x^{0}=y$ and $i,j=1,...N$, and $\varphi^{0}(\bm{x})=\mathcal{V}(\bm{x})$

[TABLE]

Here in (2.8), we can clearly see that if we set $\mathcal{V}(\bm{x})=\mathcal{V}(y)\ \Rightarrow\ \partial_{\mu}\mathcal{V}=0\ \forall\ \mu\neq 0$

[TABLE]

Thus, the only way to ensure that $\mathcal{P}^{\mu}_{\mathcal{V}}$ is restricted to only one non-zero component $\mathcal{P}^{0}_{\mathcal{V}}$ is to have vanishing cross terms in the base space and configuration space metric. ie.:

[TABLE]

Now, if $g_{\mu\nu}(\bm{x})$ is independent of $x^{0}=y$,

[TABLE]

then upon applying (2.9) and (2.10) to (2.7), we shall have:

[TABLE]

Thus, we have a field momentum that is a constant of motion. In summary, the conditions for this field momentum to be a constant of field dynamics are:

the cyclic field $\mathcal{V}$ is dependent only on the cyclic co-ordinate $x^{0}=y$, 2. 2.

the metric of the co-ordinate space $g_{\mu\nu}(\bm{x})$ is independent of $x^{0}=y$, and 3. 3.

cross-terms with cyclic co-ordinates and fields must vanish in the base and configuration space metrics (ie. $g^{\mu 0}(\bm{x})=0\ \forall\ \mu\neq 0$, $h_{i0}(\bm{\varphi})=0$).

We will next describe the generator of the field theory equivalent of Hamilton’s equations of motion.

2.2 A generator of field equations

Here, we shall introduce a function in fields and field momenta that generates the scalar field theory equivalent of Hamilton’s equations of motion. In a similar manner to Legendre’s method to define the Hamiltonian, the function shall be defined as:

[TABLE]

If we take the gradient of the generator (2.12), we can show that:

[TABLE]

Applying (2.3) to (2.13), we can see that:

[TABLE]

where upon comparing the co-efficients of $\partial_{\alpha}\varphi^{i}$ and $\partial_{\alpha}\mathcal{P}^{\mu}_{i}$, we can see that:

[TABLE]

Like the Hamiltonian for point-particle mechanics, this generator is instrumental to the process of Eisenhart lifting the scalar field space.

3 Eisenhart lift of scalar field space

So far, in [16], we have seen the Eisenhart-Riemannian lift of a field space by introducing vector fields. Here, we shall perform Eisenhart lift for field theory using a cyclic scalar field only. One must ensure that a cyclic co-ordinate wrt the co-ordinate space metric is available throughout the setup to ensure the existence of a conserved quantity that enables the lift via a cyclic field without disturbing the field equations (2.14).

3.1 Riemannian field lift

Let us suppose that we have a field Lagrangian (2.8) with the conditions (2.9) and (2.10) applied given by:

[TABLE]

The generator (2.12) in this case will be:

[TABLE]

where the conserved field momentum is:

[TABLE]

If we define a scalar field function:

[TABLE]

then the field equations (2.14) deduced from the lifted generator (3.2) are:

[TABLE]

which when applied to the lifted generator (3.2), leads us to the field Lagrangian:

[TABLE]

Thus, we can say that the Riemannian lift of the field Lagrangian (3.4) is:

[TABLE]

If a co-ordinate $x^{0}=y$ cyclic wrt the metric $g_{\mu\nu}(\bm{x})$ does not exist, we can create a new co-ordinate $y$ such that $g_{\mu 0}(\bm{x})=0\ ,\ g_{00}(\bm{x})=1$, that only the new cyclic field created for the lift will depend on. This method will be called “Double Lift”, since it expands the co-ordinate space alongside the field space. However, since the other fields are independent on $y$, it cannot be a generalization of point particle theory, since the fields in a point particle theory are dependent on the sole parameter.

3.2 Lorentzian field lift

This time, we shall consider a field Lagrangian of the form:

[TABLE]

where we have decided to set $g^{00}(\bm{x})=1,h_{\mathcal{U}\mathcal{V}}(\bm{\varphi},\mathcal{U})=1$. The generator (2.12) will be:

[TABLE]

We can write the conserved field momentum as a constant $\mathcal{P}^{0}_{\mathcal{V}}=q$ to write the generator (3.7) as:

[TABLE]

Thus, the field equations generated according to (2.14) from the lifted generator (3.7) will be:

[TABLE]

If we write $\frac{q^{2}}{2}h_{\mathcal{U}\mathcal{U}}(\bm{\varphi},\mathcal{U})=-V(\bm{\varphi},\mathcal{U})$, the regular field Lagrangian deduced from the generator (3.7) will be

[TABLE]

Thus, the field Lagrangian (3.6) deduced from lifting (3.9) can be written as

[TABLE]

Other applications of the Eisenhart-Duval lift to branes are discussed in [22] where the authors study the relativistic generalization of the Chaplygin gas in Duval’s Kaluza-Klein framework, and in [23], where Gibbons applies the Eisenhart-Duval lift to a flat Lorentzian spacetime with two times.

4 Randers-Finsler formulation for Nambu-Goto action

Randers [19] performed the first significant application of Finslerian geometry to physics, describing the action in terms of the charged massive particle’s worldline and its interaction with gauge fields to account for the influence of curvature of spacetime and gauge fields in its motion, the related mechanics of which has been discussed in detail in [13]. The Finslerian connection for massless particles was discussed by Duval [20] and Elbistan, Zhang, Dimakis, Gibbons, and Horvathy in [21]. In the former, Duval formulates a general theory of geometric optics for spinning light rays on a Finsler manifold, effectively deducing Finslerian spinoptics, while the authors in the latter study the free motion of a massive particle in a Finslerian deformation of a plane gravitational wave.

So far, we have discussed the Lagrangian for a scalar field theory written in quadratic form of a regular scalar field Lagrangian. However, they often have a geometric origin, such as the world volume spanned in the scalar field space. One such example is the Nambu-Goto action discussed in string theory where the action is directly proportional to the total surface area of a worldsheet swept out in spacetime.

If the action be written as the integral of the volume spanned in configuration space:

[TABLE]

then we can deduce the scalar field momentum from (4.1) to be:

[TABLE]

Here, we can see from applying (4.2) that a rule analogous to the Maupertuis principle for point particle mechanics exists for such geometric actions.

[TABLE]

In case of strings, we will have $n=2$, and in case of a point particle $n=1$ for which the familiar Maupertuis principle discussed in classical mechanics, easily evident for Randers-Finsler geodesics [13, 14, 15] re-emerges. One can easily see from (4.3) that for world volumes, the trace of the energy momentum tensor shall vanish, similar to the overall Hamiltonian for Randers-Finsler geodesics as shown in [13, 14].

[TABLE]

If any one of the fields $\varphi^{0}=\mathcal{V}$ is cyclic, and satisfies all the conditions mentioned for its conjugate field momentum $\mathcal{P}^{0}_{\mathcal{V}}$ to be conserved as described in Sect. 2.1, then we will have from (4.3):

[TABLE]

As we know, the last term for a field theory being a total divergence term is an additive freedom factor that can be omitted from the Lagrangian (4.5) to formulate a Jacobi Lagrangian.

[TABLE]

However, there is no constraint for a geometric world volume action in curved space, except for a point particle, so we are unable to determine a Jacobi metric as done in [13, 15]. This is evident if one uses (4.2) to write a matrix:

[TABLE]

[TABLE]

Thus, we can see that only for a point particle $n=1$ we will have the constraint, described in [13, 15].

[TABLE]

However, for the Nambu-Goto action of string, we can formulate a generator of a field equation from (4.8) using (4.2) and (4.7)

[TABLE]

[TABLE]

where in the 2nd equation of (4.10), $\Omega^{j}_{ik}$ is the connection on the scalar field space introduced in (2.6), and the equation

[TABLE]

can easily be verified by applying the Lagrangian $L=\sqrt{M}$ (all constant factors adjusted to 1) for Nambu-Goto action the Euler-Lagrange field equation (2.3). Such a generator can be formulated only for point particle and the Nambu-Goto string. Furthermore, the Randers-Finsler form for the Nambu-Goto action can intuitively be written as:

[TABLE]

where $\rho(\sigma)$ is the density along the string, and $\bm{A}(\bm{\varphi})$ are the gauge fields in scalar field space. In this case, we can define a gauge covariant field momentum

[TABLE]

which will substitute the canonical field momentum $\mathcal{P}^{\tau}_{i}$ in the formulation of the generator of equations for a string introduced previously in (4.9), as done in [13, 15].

5 Conclusion and Discussion

We started by reviewing scalar field theory, showing how a conserved quantity can exist in scalar field theory, and introduced a generator of the field theory equivalent of Hamilton’s equations of motion. Such conditions are not necessary for point particle, where non-zero cross terms in the field space metric are allowed.

We then showed how to perform both, the Riemannian and Lorentzian versions of the Eisenhart lifts, for n-dimensional scalar field theories, using the Lagrangian of the Polyakov form. While it is restricted by specific conditions compared to that described in [16], it is more similar and comparable to the procedure for point particle mechanics, and can be considered the simplest and most direct extension of the geometric lifting procedure to scalar field theory.

If a cyclic co-ordinate that the metric is independent of is unavailable, the co-ordinate space could also be expanded to include one, describing a “Double Lift”. However, a “Double Lifted” field theory cannot be considered a generalization of point particle theory since the fields are independent of the cyclic co-ordinate.

Finally, we can see that many mechanical principles associated with point particle, such as Maupertuis principle can be generalized to world volume actions, for which one setting gives the Nambu-Goto action. Furthermore, a generator of equations analogous to a Hamiltonian can be described only for strings and point particles, where in the latter case, it is constrained to a constant value.

Acknowledgement

We wish to acknowledge G. W. Gibbons, M. Cariglia, and Joydeep Chakravarty for various discussions and support, P. Horvathy, K. Morand and A. Galajinsky for supportive comments, and P. Mukhopadhay for introducing us to scalar field theory, all of which were instrumental in the preparation of this article.