Eisenhart lift and Randers-Finsler formulation for scalar field theory
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r Eisenhart lifting the Field Space with a cyclic field
Sumanto Chanda , Partha Guha International Centre for Theoretical SciencesNo. 151, Shivakote, Hesaraghatta Hobli, Bengaluru, Karnataka 560089, India. [email protected] S.N. Bose National Centre for Basic SciencesJD Block, Sector-3, Salt Lake, Calcutta-700106, India.Department of Mathematics, Khalifa UniversityAbu Dhabi, United Arab Emirates. ∗ [email protected] April 28, 2020
Abstract
We study n-dimensional scalar field theory as a generalization of point particle mechanics using thePolyakov action, and demonstrate how to extend Lorentzian and Riemannian Eisenhart lifts to the theoryin a similar and comparable manner.
The application of geometric methods in physics led to modern theories at the heart of active research, such asString Theory and Braneworld Cosmology. The Kaluza-Klein theory is one such topic, where one adds extradimensions to account for other interactions. On the other hand, the Eisenhart lift, inspired by Eisenhart [1],and further developed and studied by Gibbons, Duval, Horvathy and others [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]is a procedure that formulates a new background curvature that replaces the gauge fields that influence aparticle’s motion by adding degrees of freedom associated with symmetry.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 offreedom associated with symmetry. When a theory is potential free, the corresponding equation is tautolog-ically geodesic. Adding potentials to the theory causes a deviation from geodesic. However, by using theEisenhart lift, this can be embedded into the geodesic equation of the metric in higher-dimensional spacewith one extra fictious field.Scalar field theory in n-dimensional field space is a generalization of point particle mechanics, which leadsus to ask if we can also similarly perform an Eisenhart lift of the scalar field space. The main obstacle toperforming the Eisenhart lift on field space is the divergence term involved in the field equations. Due to ∗ Address after 1st August, 2020 his, Finn, Karamitsos and Pilaftsis have demonstrated how to execute the Riemannian lift [16] of a scalarfield space with Lagrangian of the Non-Linear Sigma Model (NLSM) by introducing vector fields.The goal of this paper is to study the Eisenhart lift of the scalar field space. We shall thoroughlydescribe both, the Riemannian and Lorentzian lifting processes, in a simple and original way to generalizethe Eisenhart lift for application on scalar field space.
While scalar field theory is a generalization of point particle mechanics, special circumstances in the latterallow a conserved quantity to exist due to an available cyclic co-ordinate, which is not normally possible in theformer. In this section, we shall review scalar field theory formulation, discuss how a conserved quantity canexist if a cyclic field is available, and introduce a generator of field equations analogous to the Hamiltonian.If we have an n -dimensional scalar field space { ϕ i ( x ) } and define the Lagrangian L = L ( g µν , ϕ i , ∂ µ ϕ i ) on thespace, the field indices 0 ≤ i, j, k, m ≤ n , and the field action: S = Z V d n x √− g L ( g µν , ϕ i , ∂ µ ϕ i ) , (2.1)then remembering that δ ( ∂ µ ϕ i ) = ∂ µ ( δϕ i ) = ∇ µ ( δϕ i ), the arbitrary variation of the action (2.1) in thespacetime region V can be expanded using chain rule: δS = Z V d n x √− g (cid:20)(cid:18) g µν L + ∂ L ∂g µν (cid:19) δg µν + (cid:26) ∂ L ∂ϕ i − ∇ µ (cid:18) ∂ L ∂ ( ∇ µ ϕ i ) (cid:19)(cid:27) δϕ i + ∇ µ (cid:18) ∂ L ∂ ( ∇ µ ϕ i ) δϕ i (cid:19)(cid:21) . (2.2)Thus, the field equation of motion for classical trajectories is given by: ∂ L ∂ϕ i − ∇ µ ( P µi ) = 0 , where P µi := ∂ L ∂ ( ∂ µ ϕ i ) (2.3)To see how classical mechanics for single point particle is a field theory as well, setting x µ → τ ⇒ d n x → dτ ,and g µν ( x ) → g → n -dimensional scalar field space, we will essentially deal with the Non-Linear Sigma Model (NLSM) Lagrangian given as: L = 12 g µν ( x ) h ij ( ϕ ) ∂ µ ϕ i ∂ ν ϕ j . (2.4)and its Euler-Lagrange field equation given by (2.3) as: g µν ( x ) (cid:16) ∇ µ ∂ ν ϕ i + Ω ijk ∂ µ ϕ j ∂ ν ϕ k (cid:17) = 0 , (2.5)where the field space connection symbol Ω ijk in (2.5) isΩ ijk = h im ( ϕ )2 (cid:18) ∂h mj ( ϕ ) ∂ϕ k + ∂h mk ( ϕ ) ∂ϕ j − ∂h jk ( ϕ ) ∂ϕ m (cid:19) . Now we shall proceed to discuss how a conserved quantity may exist in scalar field theory as a result of acyclic field. .1 Cyclic fields Even if the field Lagrangian L is independent of a cyclic field ϕ = V , the corresponding field momentum P µ V is not necessarily a conserved quantity, since according to (2.3), we have a covariant divergence equation. ∇ µ P µ V = ∂ µ P µ V + Γ µµρ P ρ V = ∂ L ∂ V = 0 . (2.6)Now consider the following field Lagrangian based on the model (2.4) where µ, ν = 0 , , ...n, x = y and i, j = 1 , ...N , and ϕ ( x ) = V ( x ) = V ( y ) L = 12 g µν ( x ) (cid:2) h ij ( ϕ ) ∂ µ ϕ i ∂ ν ϕ j + 2 h i ( ϕ ) ∂ µ ϕ i ∂ ν V + h ( ϕ ) ∂ µ V ∂ ν V (cid:3) . (2.7)Here in (2.7), we can clearly see that if we set V ( x ) = V ( y ) ⇒ ∂ µ V = 0 ∀ µ = 0 P µ V = ∂ L ∂ ( ∂ µ V ) = g µν ( x ) (cid:2) h i ( ϕ ) ∂ ν ϕ i + h ( ϕ ) ∂ ν V (cid:3) V ( x )= V ( y ) = g µν ( x ) h i ( ϕ ) ∂ ν ϕ i + g µ ( x ) h ( ϕ ) ∂ V . Thus, the only way to ensure that P µ V is restricted to only one non-zero component P V is to have vanishingcross terms in the base space and configuration space metric. ie.: g µ ( x ) = h i ( ϕ ) = 0 , V = V ( y ) ⇒ P µ V = 0 ∀ µ = 0 . (2.8)Now, if g µν ( x ) is independent of x = y , ∂ g µν = 0 ⇒ Γ µµ = 12 g ρσ ∂ g ρσ = 0 . (2.9)then upon applying (2.8) and (2.9) to (2.6), we shall have: ∇ µ P µ V = ∇ P V = ∂ P V = ∂ L ∂ V = 0 ⇒ P V = const. (2.10)Thus, we have a field momentum that is a constant of motion. In summary, the conditions for this fieldmomentum to be a constant of field dynamics are:1. the cyclic field V is dependent only on the cyclic co-ordinate x = y ,2. the metric of the co-ordinate space g µν ( x ) is independent of x = y , and3. cross-terms with cyclic co-ordinates and fields must vanish in the base and configuration space metrics(ie. g µ ( x ) = 0 ∀ µ = 0, h i ( ϕ ) = 0).We will next describe the generator of the field theory equivalent of Hamilton’s equations of motion. Here, we shall introduce a function in fields and field momenta that generates the scalar field theory equivalentof Hamilton’s equations of motion. In a similar manner to Legendre’s method to define the Hamiltonian, thefunction shall be defined as: G := P µi ∂ µ ϕ i − L . (2.11)If we take the gradient of the generator (2.11), we can show that: ∂ α G = ∂G∂ϕ i ∂ α ϕ i + ∂G∂ P µi ∂ α P µi = ∂ µ ϕ i ∂ α P µi + P µi ∂ α ∂ µ ϕ i − (cid:18) ∂ L ∂ϕ i ∂ α ϕ i + ∂ L ∂ ( ∂ µ ϕ i ) ∂ α ∂ µ ϕ i (cid:19) (2.12) pplying (2.3) to (2.12), we can see that: ∂G∂ϕ i ∂ α ϕ i + ∂G∂ P µi ∂ α P µi = ∂ µ ϕ i ∂ α P µi − ∇ µ P µi ∂ α ϕ i , where upon comparing the co-efficients of ∂ α ϕ i and ∂ α P µi , then we can see that: ∂ µ ϕ i = ∂G∂ P µi , ∇ µ P µi = − ∂G∂ϕ i . (2.13)Like the Hamiltonian for point-particle mechanics, this generator is instrumental to the process of Eisenhartlifting the 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 cyclicco-ordinate wrt the co-ordinate space metric is available throughout the setup to ensure the existence of aconserved quantity that enables the lift via a cyclic field without disturbing the field equations (2.13).Let us consider the field Lagrangian according to the Non-Linear Sigma Model (NLSM) is given by: L = 12 g µν ( x ) h ij ( ϕ , χ ) ∂ µ ϕ i ∂ ν ϕ j − V ( ϕ , χ ) , (3.1)where one of the co-ordinates x = y is cyclic wrt the metric g µν ( x ), and the field χ ( x ) = y . The fieldmomenta P µi for this Lagrangian can be defined as: P µi = ∂L∂ ( ∂ µ ϕ i ) = g µν ( x ) h ij ( ϕ , χ ) ∂ ν ϕ j . (3.2)Since ∂ µ χ = δ µ , the gradient term ∂ χ = 1 is probably concealed within the Lagrangian (3.1). Keeping thisin mind, and using (2.11) and (3.2), the generator is: G ( ϕ i , χ, P µi ) = P µχ ∂ µ χ + P µi ∂ µ ϕ i − L = P χ + 12 g µν ( x ) h ij ( ϕ , χ ) P µi P νj + V ( ϕ , χ ) . (3.3)Under the lift with P V = const : V ( ϕ , χ ) = Φ( ϕ , χ )( P V ) , P χ = P U P V , (3.4)the generator (3.3) becomes: G ( ϕ i , χ, P µi , P U , P V ) = 12 g µν ( x ) h ij ( ϕ , χ ) P µi P νj + Φ( ϕ , χ )( P V ) + P U P V . (3.5)Now we shall describe Lorentzian and Riemannian lifts for a field space. P χ = 0 ) In this case, the original Lagrangian has explicit dependence on x = y as originally defined. Thus, the liftedgenerator will be (3.5), and the field equations generated by it will be: ∂ µ ϕ i = ∂G∂ P µi = g µν ( x ) h ij ( ϕ , χ ) P νj ∂ U = ∂G∂ P U = P V = const ⇒ U ( y ) = P V y = P V χ ( y ) ∂ V = ∂G∂ P V = 2Φ( ϕ , χ ) P V + P U (3.6) hich upon application to (3.5), and switching dependence from χ ( y ) with U ( y ) = P y V χ ( y ) by writing e h ij ( ϕ , U ) = h ij ( ϕ , χ ) , e Φ( ϕ , U ) = Φ( ϕ , χ ), where U ( y ) = P y V χ ( y ) will give us: L = 12 g µν ( x ) e h ij ( ϕ , U ) ∂ µ ϕ i ∂ ν ϕ j − e Φ( ϕ , U )( ∂ U ) + ( ∂ U )( ∂ V ) ≡ g µν ( x ) H IJ ( ξ ) ∂ µ ξ I ∂ ν ξ J , (3.7)where { ξ ( x ) } = { ϕ ( x ) , U ( y ) , V ( y ) } , I, J = 0 , , ...N + 1, and H ij ( ξ ) = e h ij ( ϕ , U ), H ( ξ ) = − e Φ( ϕ , U ), H N +1 ( ξ ) = 1. P χ = 0 ) In this case, we are considering that the Lagrangian is not explicitly dependent on the cyclic co-ordinate y (ie. h ij ( ϕ , y ) = h ij ( ϕ )), the generator (3.5) will read as: G ( ϕ i , P µi , P V ) = 12 g µν ( x ) h ij ( ϕ ) P µi P νj + Φ( ϕ )( P V ) . (3.8)The field equations deduced from the lifted generator (3.5) according to (2.13) are: ∂ µ ϕ i = ∂G∂ P µi = g µν ( x ) h ij ( ϕ ) P νj , ∂ V = ∂G∂ P V = 2Φ( ϕ ) P V (3.9)which when applied to the lifted generator (3.8), gives us the field Lagrangian: L = 12 g µν ( x ) h ij ( ϕ ) ∂ µ ϕ i ∂ ν ϕ j + 14Φ( ϕ ) ( ∂ V ) ≡ g µν ( x ) H IJ ( ξ ) ∂ µ ξ I ∂ ν ξ J , (3.10)where { ξ ( x ) } = { ϕ ( x ) , V ( y ) } , I, J = 0 , , ...N , and H ij ( ξ ) = h ij ( ϕ ), H ( ξ ) = (4Φ( ϕ )) − .If a co-ordinate x = y cyclic wrt the metric g µν ( x ) does not exist, we can create a new co-ordinate y suchthat g µ ( x ) = 0 , g ( x ) = 1, that only the new cyclic field created for the lift will depend on. This methodwill be called “Double Lift”, since it expands the co-ordinate space alongside the field space. However, sincethe other fields are independent on y , it cannot be a generalization of point particle theory, since the fieldsin a point particle theory are dependent on the sole parameter. 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 conditionsare 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 Non-Linear Sigma Model (NLSM). Whileit is restricted by specific conditions compared to that described in [16], it is more similar and comparableto the procedure for point particle mechanics, and can be considered the simplest extension of the geometriclifting procedure to scalar field theory.If a cyclic co-ordinate that the metric is independent of is unavailable, the co-ordinate space could alsobe expanded to include one, describing a “Double Lift”. However, a “Double Lifted” field theory cannot beconsidered a generalization of point particle theory since the fields are independent of the cyclic co-ordinate.
Acknowledgement
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