aa r X i v : . [ h e p - t h ] J a n Stochastic Quantization on Lorentzian Manifolds
Folkert Kuipers ∗ Department of Physics and Astronomy, University of Sussex,Brighton, BN1 9QH, United Kingdom
February 1, 2021
Abstract
We embed Nelson’s stochastic quantization in the Schwartz-Meyer second order geometryframework. The result is a non-perturbative theory of quantum mechanics on (pseudo)-Riemannian manifolds. Within this approach, we derive stochastic differential equations formassive spin-0 test particles charged under scalar potentials, vector potentials and gravity.Furthermore, we derive the associated Schr¨odinger equation. The resulting equations showthat massive scalar particles must be conformally coupled to gravity in a theory of quantumgravity. We conclude with a discussion of some prospects of the stochastic framework. ∗ E-mail: [email protected] ontents k -forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Exterior derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Interior products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8 Lie derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 Embeddings into higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Introduction
The construction of a theory of quantum gravity is one of the main open issues in theoreticalhigh energy physics. One of the reasons why such a theory is desirable is that general relativityis unable to completely describe physical aspects of gravity at extremely high energy scales.This feature is most prominent in the fact that singularities seem to be unavoidable in generalrelativity, when natural assumptions are made [1–4].From a physical perspective, the formation of such singularities would require the continuouscollapse of a matter distribution to a delta distribution located at the singularity. On R n one canmake sense of such a collapse, as one can construct a family of smooth distributions that convergesto the delta distribution. In general relativity, on the other hand, point-like sources cannot beobtained as a continuous limit of matter distributions defined on manifolds with smooth metrics,as the Einstein equations must be satisfied during the collapse [5].It is expected that this paradox will be resolved, when general relativity is embedded into aquantum theory such that gravity is quantized. However, when one attempts such an embeddingusing standard quantum field theory methods, one runs into the problem that the resultingquantum theory is non-renormalizable [6]. Up to the Planck scale, one can still make predictionsregarding quantum gravity using effective field theory methods, since the ultra-violet divergencesresponsible for the non-renormalizability of the theory can be kept under control perturbatively.However, beyond the Planck scale this is no longer true, which renders the theory incomplete.Over the last decades many approaches to an ultra-violet complete theory of quantum gravityhave been developed, and many interesting insights have been obtained within these approaches.In this paper, we argue that Nelson’s stochastic quantization framework could help gain furtherinsight in theories of quantum gravity. We will motivate this by showing that stochastic quan-tization allows to construct a well defined non-perturbative theory of quantum mechanics on(pseudo-)Riemannian manifolds.We will adopt the framework of stochastic quantization that was proposed by F´enyes [7] andKershaw [8], rederived by Nelson [9–11] and further developed by many others. The main ideagoverning stochastic quantization is that quantum mechanics can be derived from a stochastictheory. In this more fundamental theory all particles follow trajectories through a randomlyfluctuating background field. Due to the interactions with this background field all matterbehaves quantum mechanically. An equivalent way to state this idea is that all particles andfields are defined on a randomly fluctuating space-time.We focus in this paper on ordinary quantum mechanics. We will thus work with point-likeparticles instead of fields. Moreover, we work on a fixed Lorentzian manifold. Therefore, themetric is not considered to be a dynamical field. We leave extensions to a field theory frameworkand to dynamical geometries for future work. In the stochastic quantization framework suchextensions lead to a theory of quantum gravity. Since the quantization procedure in stochastic quantization is very different from more commonlyused quantization procedures, we will compare the main steps to canonical quantization. In acanonical quantization procedure one starts with a classical Hamiltonian H ( p, x ) and promotesthe variables p, x to operators P, X such that H ( p, x ) → ˆ H ( P, X ) . One could call this a ‘passive’ description of stochastic quantization, since the space-time fluctuates, while inthe previous ‘active’ description the matter defined on the space-time fluctuates. P µ , X ν ] = i ~ δ νµ , [ P µ , P ν ] = 0 , [ X µ , X ν ] = 0 . (1.1)Moreover, one postulates the existence of a wave function Ψ, which is an element of a complexHilbert space with L -norm, that can be used to calculate observables, i.e., h Ψ | ˆ O | Ψ i = O. (1.2)In stochastic quantization, one starts with a classical Lagrangian L c ( x, v, τ ), and promotesthe position of a particle x to a stochastic process X ( τ ). Since the stochastic process is notdifferentiable, one can define two velocities v ± using conditional expectations: v + ( X ( τ ) , τ ) = lim h ↓ E [ X ( τ + h ) − X ( τ ) | X ( τ )] ,v − ( X ( τ ) , τ ) = lim h ↓ E [ X ( τ ) − X ( τ − h ) | X ( τ )] . (1.3)One can then introduce a stochastic Lagrangian L c ( x, v, τ ) → L ( X, V + , V − , τ ) = 12 [ L c ( X, V + , τ ) + L c ( X, V − , τ )] (1.4)Moreover, one fixes the quadratic variation of the process X by the background hypothesis:[[ X µ , X ν ]]( τ ) = ~ m δ νµ τ. (1.5)We remind the reader that the joint quadratic variation of two processes X, Y is itself a stochasticprocess and can be written as[[
X, Y ]]( τ ) = X ( τ ) Y ( τ ) − X (0) Y (0) − Z τ X ( s ) dY ( s ) − Z τ Y ( s ) dX ( s ) . (1.6)The Itˆo integral used in this expression is defined by Z τ f τ i f ( X, τ ) dX := lim k →∞ X [ τ j ,τ j +1 ] ∈ π k f ( X ( τ j ) , τ j ) [ X ( τ j +1 ) − X ( τ j )] , (1.7)where π k is a partition of [ τ i , τ f ].Observables in stochastic quantization can be calculated using the expectation E , which isdefined on a filtered probability space, and evaluated as a Lebesgue integral in the L -space ofstochastic processes. The construction of expectation values in modern probability theory asfounded by Kolmogorov [12] requires the existence of a probability measure P in the probabilityspace, and a measure µ in the L -space, but not the existence of a probability density. Therefore,the wave function Ψ no longer needs to be postulated in stochastic quantization. More commonly used notations for d [[ X i , X j ]] are d [ X i , X j ] or dX i dX j . We use the double brackets instead toavoid confusion with the commutator, first order bilinear tensors and second order vectors that will be introducedin section 2. If a probability density ρ ( x ) exists, one has the familiar relation dµ ( x ) = ρ ( x ) d n x . will become stochas-tic and comparable to a frictionless Brownian motion. This Brownian motion is imposed to betime-reversible. This additional assumption introduces an important distinction from Brownianmotion processes that are more commonly studied in statistical physics.Most stochastic diffusion processes that are studied in physics, such as for example theOrnstein-Uhlenbeck process, are dissipative diffusions. These processes are not time reversible,and energy is transferred from the system to the environment until an equilibrium is reached.The processes studied in stochastic mechanics are conservative diffusion processes. These pro-cesses are time-reversible and the expected energy transfer between the system and environmentis 0 at all times.The fact that the wave function is no longer fundamental in stochastic quantization has twofurther important consequences. First, constructing normalized wave functions on Riemannianmanifolds is a difficult task, that complicates extensions of ordinary quantum mechanics tomanifolds. This problem is circumvented in the stochastic approach, as the wave function nolonger needs to exist globally.Secondly, due to the secondary role of the wave function, there is no measurement problem instochastic mechanics. The wave function and probability density in stochastic mechanics have thesame status as in standard probability theory. A theoretically perfect measurement in stochasticmechanics thus corresponds to conditioning of the process. Conditioning is a mathematicaloperation that still leads to collapse of the wave function, but since the wave function is onlya mathematical construct and not a physical object, this does not correspond to a physicalinteraction.
The success of stochastic quantization relies on the relation between probability density functionsassociated to stochastic processes and partial differential equations. In the case of dissipativediffusions, the probability density associated to the solution of a stochastic differential equationevolves according to a parabolic differential equation. This result is known as the Feynman-Kacformula [14]. An example of this relation is the fact that the probability density of a dissipativeBrownian motion evolves according to the heat equation, which is a real diffusion equation.A similar relation exists for conservative diffusion processes. For example, the probabilitydensity of a conservative Brownian motion evolves according to the Schr¨odinger equation, whichis a complex diffusion equation. This result is closely related to the Feynman-Itˆo formula [15,16].Before this latter relation was formally established, it was discovered independently by F´enyes,Kershaw and Nelson [7–11] that the Schr¨odinger equation can be derived from a stochastic theory,if one assumes that particles follow a time-reversible stochastic process, governed by a stochasticversion of Newton’s second law, where the force is derived from a potential.The theory that was developed in this way is called stochastic mechanics. The immediateconsequence of this discovery is that all predictions of quantum mechanics that follow from theSchr¨odinger equation, are also predictions of stochastic mechanics. Later it was shown that the Stochastic quantization has yet to be extended to massless particles. Notice that eq. (1.5) characterizes a scaled Brownian motion [13]. Despite the successes described above, stochastic quantization has never been widely studied.We will therefore review some of the main concerns that have been raised against stochasticquantization.Historically, one of the more prominent confusions arose from the idea that a diffusion processis necessarily dissipative, and cannot give rise to quantum mechanics. As argued before, this isnot the case, when the diffusion is time-reversible. This point has been well explained by Nelsonin section 14 of Ref. [11], where an analogy is made with the difference between Aristoteleanand Galilean dynamics. It should be noted that in order to describe entanglement in stochasticquantization, the background field has to be non-local. This particular feature was dislikedby Nelson, cf. e.g. Ref. [44]. We stress that this non-locality is merely a feature of quantummechanics, and not specific to stochastic quantization. Moreover, it is an open question, whetherthe non-locality of the background can be avoided, if one considers non-Markovian diffusionprocesses.Another concern that may be raised against stochastic mechanics is that it can be regardedas a hidden variable theory, as it is assumed that a background field exists that is responsiblefor the quantum fluctuations. One could thus expect that stochastic mechanics violates the Bellinequalities. We will avoid this issue by assuming that the background field is fundamentallystochastic, in the sense that the fluctuations cannot be derived from a more fundamental theory.This is different from for example the Brownian motion of a colloid suspended in a liquid, wherethe trajectory of the colloid can theoretically be derived by solving the equation of motion of allthe molecules in the liquid.A more pressing issue for stochastic quantization is Wallstrom’s criticism [45,46], which states Let us be a bit more precise, as the process is slightly more complicated in stochastic quantization: afterpassing through one of the slits, the particle will diffuse according to a one slit diffusion process. However, dueto the imposed time-reversibility of the motion, it will transition into a double slit diffusion process. The lengthscale associated to this transition is the width of the slit, cf. e.g. sections 16 and 17 in Ref. [11]. π periodicity of the wave function has to be imposed as an additional assumption. Suchan assumption must be made ad hoc, since the wave function is not a fundamental object inthe theory. Several responses against this criticism have been given, such as for example theincorporation of zitterbewegung [47, 48], adding a postulate regarding the boundedness of theLaplace operator acting on the probability density [49] or by adding the assumption of unitarityof superpositions of wave functions [34]. It is also worth mentioning that on nodal manifoldswinding numbers lead to a periodicity factor in the wave function [11], which could resolveWallstrom’s criticism.Since no consensus yet exists about the solution of Wallstrom’s criticism, we will take a morepragmatic approach: we accept this ad hoc constraint and remain agnostic about its solution.The reason for this is that imposing such a constraint is only problematic at a foundational level.Even if Wallstrom’s criticism cannot be resolved within stochastic quantization, the theory canstill be used as an alternative mathematical model of quantum theory, and can thus be used tomake predictions about quantum systems. As we will show in this paper, a particular advantageof the stochastic model is that it can be formulated on (pseudo-)Riemannian manifolds, whichcould help guide the way towards a theory of quantum gravity.A more practical concern regarding stochastic quantization is that analytical calculationsrequire to solve stochastic differential equations. This is notoriously difficult. In fact, an im-portant solution method relies on the mapping stochastic differential equations to path integralproblems and to partial differential equations, as established by the Feynman-Kac formula. It isthus expected that many calculations can more easily be performed in ordinary quantum theory.This would render stochastic mechanics as an alternative mathematical model unnecessary. De-spite this fact, it is expected that stochastic quantization could prove to be useful in numericalcalculations, and a small number of analytical calculations. More interesting, however, is the po-tential of stochastic quantization on a more formal level. In particular, it could prove to be usefulin mathematically rigorous definitions of the path integral, which is expected to be essential forconstructing a theory of quantum gravity. We note here that stochastic approaches already serveas one of the stepping stones of the Euclidean approach in quantum field theory [50–53]. Before moving on, let us summarize the fundamental assumptions of stochastic quantization:we assume that all particles follow well defined trajectories through a diffeomorphism invariantbackground field. This background field induces stochastic fluctuations such that the motion ofparticles resembles a conservative Brownian motion. Moreover, the quadratic variation of thisprocess scales with the Planck constant according to the background hypothesis. We have thefollowing postulates: • All observables are invariant under a change of coordinate system. • The stochastic motion of a particle with mass m is Markovian. • The stochastic motion of a particle with mass m is time-reversible. • The stochastic motion obeys the structure equation [[ X µ , X ν ]]( τ ) = ~ m δ νµ τ .We note that the classical limit of the theory can be obtained straightforwardly by taking thelimit ~ →
0. 6 .5 Main results of the paper
In this paper, we work in the ( − + ++) signature with a Riemann tensor defined by R ρσµν = ∂ µ Γ ρνσ − ∂ ν Γ ρµσ + Γ ρµκ Γ κνσ − Γ ρνκ Γ κµσ and Ricci tensor R µν = R ρµρν . In addition, we set c = 1throughout the paper.The main result we present in this paper is the following: in the stochastic quantizationframework, a massive scalar particle moving on a Lorentzian manifold and governed by thestochastic Lagrangian L ( X, V + , V − , τ ) = 12 L c ( X, V + , τ ) + 12 L c ( X, V − , τ ) (1.8)where the classical Lagrangian is given by L c ( x, v, τ ) = m g µν ( x ) v µ v ν − ~ A µ ( x, τ ) v µ − U ( x, τ ) (1.9)with x = ( t, ~x ) and τ is the proper time, evolves according to the Stratonovich stochastic differ-ential equation m g µν (cid:0) d X ν + Γ νρσ dX ρ dX σ (cid:1) = (cid:18) ~ ∂ τ A µ − ∇ µ U − ~ m ∇ µ R (cid:19) dτ − ~ ( ∇ µ A ν − ∇ ν A µ ) dX ν dτ. (1.10)Furthermore, if the probability density ρ ( x, τ ) associated to the probability measure µ = P ◦ X − exists, one can construct the wave functionΨ( x, τ ) = p ρ ( x, τ ) exp (cid:26) i ~ E (cid:20)Z ττ i L (cid:0) X ( t ) , V + ( t ) , V − ( t ) , t (cid:1) dt (cid:12)(cid:12)(cid:12) X ( τ ) = x (cid:21)(cid:27) (1.11)that evolves according to a generalization of the Schr¨odinger equation given by i ~ ∂∂τ Ψ = (cid:20) − ~ m (cid:18)h ∇ µ + iA µ ih ∇ µ + iA µ i − R (cid:19) + U (cid:21) Ψ . (1.12)This wave function obeys the Born rule | Ψ( x, τ ) | = ρ ( x, τ ) . (1.13)If there is no explicit proper time dependence in A µ or U , one can solve by separation of variablessuch that Ψ( x, τ ) = X k φ k ( x ) exp (cid:18) i m λ k ~ τ (cid:19) , (1.14)where φ k ( x ) solves the generalization of the Klein-Gordon equation given by ~ (cid:18)h ∇ µ + iA µ ih ∇ µ + iA µ i − R (cid:19) φ k = m λ k φ k + 2 m U φ k . (1.15)We note that the derivation of eqs. (1.11), (1.12) and (1.13) is a well established result on R n , see e.g. Refs. [8–11,19,28]. Moreover, partial extensions to Riemannian manifolds have beenknown for some time, cf. Refs. [11, 20–24].In this paper, we show that these results can be generalized to pseudo-Riemannian mani-folds. An important ingredient for these extensions is the second order geometry as developed7y Schwartz and Meyer [54–56]. This is an extension of ordinary differential geometry that allowsto describe stochastic processes on manifolds. In addition to the extension of stochastic quan-tization to pseudo-Riemannian manifolds, we will give some new interpretations of stochasticquantization.This paper is organized as follows: in the next section, we review second order geometry; insection 3, we introduce the relevant semi-martingale processes for quantum mechanics; section4 discusses integration along semi-martingales on manifolds; in section 5, we discuss stochasticvariational calculus; in section 6, we discuss the shape of the stochastic action; in section 7, weput everything together and derive the stochastic differential equations for quantum mechanicalscalar test particles on pseudo-Riemannian manifolds, and the associated Schr¨odinger equation.Finally, in section 8, we conclude and summarize some future perspectives of the stochasticapproach. In this section, we review the theory of Schwartz-Meyer second order geometry, that can be usedto extend the theory of stochastic calculus to manifolds. The first three subsections are looselybased on Ref. [56]. The later subsections contain new material and extend some importantconcepts from first order geometry into second order geometry. For more detail we refer to thework of Emery [56] and the original works by Schwartz [54] and Meyer [55].
We consider a ( n = d + 1)-dimensional pseudo-Riemannian manifold M with the usual firstorder tangent and cotangent spaces T x M , T ∗ x M . For every x ∈ M and any coordinate chartcontaining x one can write down bases for the tangent and cotangent space respectively givenby { ∂ µ | µ ∈ { , , ..., d }} and { dx µ | µ ∈ { , , ..., d }} . In particular for v ∈ T x M and ω ∈ T ∗ x M wehave v = v µ ∂ µ ,ω = ω µ dx µ . (2.1)Furthermore, a form ω ∈ T ∗ x M can often be written as the differential form of some function f : M → R i.e. ω = df = ∂ µ f dx µ . (2.2)The product rule for such differential forms is given by d ( f g ) = f dg + g df. (2.3)In addition, there exists a metric associated to the tangent space that is given by g : T x M ⊗ T x M → R s . t . ( v, w )
7→ h v | w i = g µν v µ w ν , (2.4)and is bilinear, symmetric and non-degenerate. Moreover the metric induces an isomorphism g Z : T x M → T ∗ x M between the tangent and cotangent space, that is defined by h g Z ( v ) , w i = h v, g Z ( w ) i = h v | w i (2.5)We define a similar bracket for two forms α, β ∈ T ∗ x M by h α | β i = h α, g \ ( β ) i = h g \ ( α ) , β i . (2.6)8e will now define a second order tangent space and cotangent space ˜ T x M , ˜ T ∗ x M . For every x ∈ M and any coordinate chart containing x one can write down bases for the tangent andcotangent space respectively given by (cid:8) ∂ µ , ∂ µν (cid:12)(cid:12) µ ≤ ν ∈ { , , ..., d } (cid:9) and { d x µ , dx µ · dx ν | µ ≤ ν ∈ { , , ..., d }} . In particular, for V ∈ ˜ T x M and Ω ∈ ˜ T ∗ x M we have V = v µ ∂ µ + v µν ∂ µν , Ω = ω µ d x µ + ω µν dx µ · dx ν . (2.7)Notice that T x M ⊂ ˜ T x M and T ∗ x M ⊂ ˜ T ∗ x M . Furthermore, ∂ µν := ∂ µ ∂ ν is a symmetric object,which implies that v µν must be symmetric. Moreover, we choose the basis of the cotangent spacedual to the basis of the tangent space. This imposes dx µ · dx ν , and ω µν to be symmetric as well.We have a duality pairing between the bases of the tangent and cotangent space such that: h ∂ µ , d x ρ i = δ ρµ , h ∂ µ , dx ρ · dx σ i = 0 , h ∂ µν , dx ρ i = 0 , h ∂ µν , dx ρ · dx σ i = 12 (cid:0) δ ρµ δ σν + δ σµ δ ρν (cid:1) . (2.8)The duality pairing of an arbitrary vector and covector is then given by h V, Ω i = v µ ω µ + v µν ω µν . (2.9)As in the classical case, forms Ω ∈ ˜ T ∗ x M can often be written as a differential form of somefunction f : M → R : Ω = d f = ∂ µ f d x µ + ∂ µν f dx µ · dx ν . (2.10)The product rule for differential forms is given by d ( f g ) = f d g + g d f + 2 df · dg (2.11)where the product of first order forms ω, θ ∈ T x M is defined by ω · θ := 12 ( ω µ θ ν + ω ν θ µ ) dx µ · dx ν = ω µ θ ν dx µ · dx ν . (2.12)Therefore, the product for two first order differential forms can be written as df · dg = ∂ µ f ∂ ν g dx µ · dx ν . (2.13)It will be useful to define mappings between the first order and second order tangent spaces.The projection map can be defined as: P : ˜ T ∗ x M → T ∗ x M s.t. ( P ( d f ) = df, P ( ω · θ ) = 0 . (2.14) Notice that dx µ · dx ν = dx µ ⊗ dx µ . More generally, one often defines the carr´e du champ operator or the squared field operator associated to alinear mapping L for two functions f, g by Γ( f, g ) := [ L ( fg ) − f Lg − g Lf ]. Cf. e.g. Lemma 6.1 in [56]. We canthen interpret df · dg as the squared field operator associated to the second order differential operator d actingon f, g . In Ref. [56] this map is called the restriction R . H from bilinear first orderforms to second order forms, such that P ◦ H = 0, given by H : T ∗ x M ⊗ T ∗ x M → ˜ T ∗ x M s . t . ( ω, θ ) ω · θ, (2.15)The adjoint of this map is denoted by H ∗ : ˜ T x M → T x M ⊗ T x M . In addition there existsa unique linear map d : T ∗ x M → ˜ T ∗ x M such that for any f ∈ C ∞ ( M , R ), ω ∈ T ∗ x M and u, v ∈ T x M d ( df ) = d f,d ( f ω ) = f dω + df · ω, h dω, [ u, v ] i = h ω, [[ u, v ]] i , h dω, { u, v }i = u h ω, v i + v h ω, u i , (2.16)where [ u, v ] is the commutator, { u, v } the anti-commutator and [[ u, v ]] the joint quadratic vari-ation of u and v .Finally, one can define maps F : ˜ T x M → T x M and G : T ∗ x M → ˜ T ∗ x M such that for anyaffine connection Γ : X ( M ) × X ( M ) → X ( M ) the following relations define a bijection between F and Γ ( F V ) f = V f − hH Γ ∗ ( df ) , V i , Γ( u, v ) f = u v f − F ( u v ) f, (2.17)where V is a second order vector and u, v are first order vector fields. A bijection between G andΓ is then defined by G ( df ) = d f − H Γ ∗ ( df ) , Γ( u, v ) f = u v f − hG ( df ) , u v i . (2.18)Moreover, F and G are each others adjoint. In this section, we investigate the change of vectors and covectors under coordinate transforma-tions. For a vector field V we find: V f = ( v µ ∂ µ + v µν ∂ µν ) f = (cid:18) v µ ∂ ˜ x ρ ∂x µ ˜ ∂ ρ + v µν ∂ µ (cid:20) ∂ ˜ x ρ ∂x ν ˜ ∂ ρ (cid:21)(cid:19) f = (cid:18) v µ ∂ ˜ x ρ ∂x µ ˜ ∂ ρ + v µν ∂ ˜ x ρ ∂x µ ∂x ν ˜ ∂ ρ + v µν ∂ ˜ x σ ∂x µ ∂ ˜ x ρ ∂x ν ˜ ∂ σρ (cid:19) f. (2.19) cf. Proposition 6.13 in Ref. [56]. cf. Theorem 7.1 in Ref. [56]. We use an underlined d to avoid confusion with the exterior derivative. cf. Proposition 7.28 in Ref. [56] X ( M ) is the space of all smooth vector fields on M , i.e. the space of all smooth sections of the tangent bundle T M . It is possible to take a connection in the defining relation for G that is different from F . If such a choice ismade, F and G are no longer each others adjoint. In this paper we will not make such a choice, as we will restrictourselves to the Levi-Civita connection. v µ → ˜ v µ = v ρ ∂ ˜ x µ ∂x ρ + v ρσ ∂ ˜ x µ ∂x ρ ∂x σ ,v µν → ˜ v µν = v ρσ ∂ ˜ x µ ∂x ρ ∂ ˜ x ν ∂x σ , (2.20)or equivalently the passive transformation laws ∂ µ → ˜ ∂ µ = ∂x ρ ∂ ˜ x µ ∂ ρ ,∂ µν → ˜ ∂ µν = ∂ x ρ ∂ ˜ x µ ∂ ˜ x ν ∂ ρ + ∂x σ ∂ ˜ x µ ∂x ρ ∂ ˜ x ν ∂ ρσ . (2.21)A form Ω transforms asΩ( V f ) = ( ω µ d x µ + ω µν dx µ · dx ν ) ( V f )= (cid:18) ω µ ∂x µ ∂ ˜ x ρ d ˜ x ρ + ω µ ∂ x µ ∂ ˜ x ρ ∂ ˜ x σ d ˜ x ρ · d ˜ x σ + ω µν ∂x µ ∂ ˜ x ρ ∂x ν ∂ ˜ x σ d ˜ x ρ · d ˜ x σ (cid:19) ( V f ) . (2.22)Therefore, the active transformation laws are given by ω µ → ˜ ω µ = ω ρ ∂x ρ ∂ ˜ x µ ,ω µν → ˜ ω µν = ω ρ ∂ x ρ ∂ ˜ x µ ∂ ˜ x ν + ω ρσ ∂x ρ ∂ ˜ x µ ∂x σ ∂ ˜ x ν , (2.23)and the passive transformation law is d x µ → d ˜ x µ = ∂ ˜ x µ ∂x ρ d x ρ + ∂ ˜ x µ ∂x ρ ∂x σ dx ρ · dx σ ,dx µ · dx ν → d ˜ x µ · d ˜ x ν = ∂ ˜ x µ ∂x ρ ∂ ˜ x ν ∂x σ dx ρ · dx σ . (2.24)The transformation laws should leave the duality pairing (2.9) invariant. Indeed we find h V, Ω i = v µ ω µ + v µν ω µν = ˜ v ρ ∂x µ ∂ ˜ x ρ ∂ ˜ x σ ∂x µ ˜ ω σ + ˜ v ρσ ∂ x µ ∂ ˜ x ρ ∂ ˜ x σ ∂ ˜ x κ ∂x µ ˜ ω κ + ˜ v ρσ ∂x µ ∂ ˜ x ρ ∂x ν ∂ ˜ x σ ∂ ˜ x κ ∂x µ ∂x ν ˜ ω κ + ˜ v ρσ ∂x µ ∂ ˜ x ρ ∂x ν ∂ ˜ x σ ∂ ˜ x κ ∂x µ ∂ ˜ x λ ∂x ν ˜ ω κλ = ˜ v µ ˜ ω µ + ˜ v µν ˜ ω µν . (2.25) In previous subsection, we found that vectors and forms in second order geometry transform inan affine but not contravariant/covariant way. This can be fixed by introducing a covariant basis { ˆ ∂ µ , ˆ ∂ µν } for ˜ T x M such that V = ˆ v µρσ ˆ ∂ µρσ = ˆ v µ ˆ ∂ µ + ˆ v νρ ˆ ∂ νρ , (2.26) One can use the Christoffel symbols to make the second term in the second line vanish. ∂ µ := ∂ µ , ˆ ∂ µν := ∂ µν − Γ ρµν ∂ ρ , ˆ v µ := v µ + v ρσ Γ µρσ , ˆ v µν := v µν . (2.27)In a similar way, we can introduce a contravariant basis for the cotangent space ˜ T ∗ x M , such thatΩ = ˆ ω µνρ d ˆ x µνρ = ˆ ω µ d ˆ x µ + ˆ ω νρ d ˆ x ν · d ˆ x ρ (2.28)with d ˆ x µ := d x µ + Γ µνρ dx ν · dx ρ ,d ˆ x µ · d ˆ x ν := dx µ · dx ν , ˆ ω µ := ω µ , ˆ ω µν := ω µν − ω ρ Γ ρµν . (2.29)It is possible to extend the notion of vector fields and forms to arbitrary ( k, l )-tensor fields.Indeed, one can construct mappings T : ( ˜ T M ) ⊗ k ⊗ ( ˜ T ∗ M ) ⊗ l → R . (2.30)In local coordinates such a tensor will be given by T = T ( µνρ ) ... ( µνρ ) k ( σκλ ) ... ( σκλ ) l ∂ ( µνρ ) ⊗ ... ⊗ ∂ ( µνρ ) k ⊗ d x ( σκλ ) ⊗ ... ⊗ d x ( σκλ ) l = T µ ...µ k σ ...σ l ∂ µ ⊗ ... ⊗ ∂ µ k ⊗ d x σ ⊗ ... ⊗ d x σ l + T ( νρ ) µ ...µ k σ ...σ l ∂ ν ρ ⊗ ∂ µ ⊗ ... ⊗ ∂ µ k ⊗ d x σ ⊗ ... ⊗ d x σ l + T µ ( νρ ) µ ...µ k σ ...σ l ∂ µ ⊗ ∂ ν ρ ⊗ ∂ µ ⊗ ... ⊗ ∂ µ k ⊗ d x σ ⊗ ... ⊗ d x σ l + ... + T µ ...µ k σ ...σ l − ( κλ ) l ∂ µ ⊗ ... ⊗ ∂ µ k ⊗ d x σ ⊗ ... ⊗ d x σ l − ⊗ dx κ l · dx λ l + T ( νρ ) ( νρ ) µ ...µ k σ ...σ l ∂ ν ρ ⊗ ∂ ν ρ ⊗ ∂ µ ⊗ ... ⊗ ∂ µ k ⊗ d x σ ⊗ ... ⊗ d x σ l + ... + T ( νρ ) ... ( νρ ) k ( κλ ) ... ( κλ ) l ∂ ν ρ ⊗ ... ⊗ ∂ ν k ρ k ⊗ dx κ · dx λ ⊗ ... ⊗ dx κ l · dx λ l . (2.31)The components of T do not transform in a covariant/contravariant way. However, one canconstruct a representation with components ˆ T such that T = ˆ T ( µνρ ) ... ( µνρ ) k ( σκλ ) ... ( σκλ ) l ˆ ∂ ( µνρ ) ⊗ ... ⊗ ˆ ∂ ( µνρ ) k ⊗ d ˆ x ( σκλ ) ⊗ ... ⊗ d ˆ x ( σκλ ) l . (2.32)If expanded as in eq. (2.32), the coefficients ˆ T do transform covariantly/contravariantly. Therelation between components T and ˆ T for a general ( k, l )-tensor can then be derived from thetransformation laws for (1 , , F , G , H . For V ∈ ˜ T x M we have F ( V ) = (cid:0) v µ + v ρσ Γ µρσ (cid:1) ∂ µ = ˆ v µ ˆ ∂ µ , (2.33) H ∗ ( V ) = v µν ∂ µ ⊗ ∂ ν = ˆ v µν ˆ ∂ µ ⊗ ˆ ∂ ν , (2.34)and for α, β ∈ T ∗ x M G ( α ) = α µ (cid:0) d x µ + Γ µρσ dx ρ · dx σ (cid:1) = ˆ α µ d ˆ x µ , (2.35) H ( α ⊗ β ) = α µ β ν dx µ · dx ν = ˆ α µ ˆ β ν d ˆ x µ · d ˆ x ν . (2.36)Therefore, all second order vectors and forms can be decomposed into first order vectors, formsand symmetric bilinear tensor products of first order vectors and forms. More generally, anysecond order ( k, l )-tensor can be decomposed into first order tensors of degree ( κ, λ ) with k ≤ κ ≤ k and l ≤ λ ≤ l . In this subsection, we extend the notion of a metric to the second order geometry framework.We can define a symmetric bilinear function ˜ g : ˜ T x M ⊗ ˜ T x M → R , that we call the secondorder metric tensor. Analogously to the first order metric, it acts on two second order vectors V, W ∈ ˜ T x M , such that ˜ g ( V, W ) = h V | W i . (2.37)Moreover, it induces an isomorphism between vectors and forms˜ g Z : ˜ T x M → ˜ T ∗ x M s . t . ( h V | W i = h ˜ g Z ( V ) , W i , h Ω , Θ i = h Ω | ˜ g \ (Θ) i . (2.38)In a local coordinate chart the metric tensor ˜ g can be written as˜ g = ˜ g ( µρσ ) ( νκλ ) d x ( µρσ ) ⊗ d x ( νκλ ) = ˜ g µν d x µ ⊗ d x ν + ˜ g µ ( κλ ) d x µ ⊗ dx κ · dx λ + ˜ g ( ρσ ) ν dx ρ · dx σ ⊗ d x ν + ˜ g ( ρσ )( κλ ) dx ρ · dx σ ⊗ dx κ · dx λ . (2.39)Using the defining isomorphism (2.38) and the duality pairing, eq. (2.9), we find the rules fortransforming second order vectors into second order forms:˜ g ( µρσ ) ( νκλ ) v νκλ = v µρσ , ˜ g µν v ν + ˜ g µ ( κλ ) v κλ = v µ , ˜ g ( ρσ ) ν v ν + ˜ g ( ρσ )( κλ ) v κλ = v ρσ . (2.40)13urthermore, the inverse ˜ g − can be used to transform second order forms into second ordervectors: ˜ g ( µρσ ) ( νκλ ) ω νκλ = ω µρσ , ˜ g µν ω ν + ˜ g µ ( κλ ) ω κλ = ω µ , ˜ g ( ρσ ) ν ω ν + ˜ g ( ρσ )( κλ ) ω κλ = ω ρσ . (2.41)The components of the metric tensor do not transform covariantly. Therefore, we define acovariant representation of the second order metric:˜ g = ˜ g ( µρσ ) ( νκλ ) d x µρσ ⊗ d x νκλ = ˜ g µν d ˆ x µ ⊗ d ˆ x ν + (cid:0) ˜ g µ ( κλ ) − ˜ g µν Γ νκλ (cid:1) d ˆ x µ ⊗ d ˆ x κ · d ˆ x λ + (cid:0) ˜ g ( ρσ ) ν − ˜ g µν Γ µρσ (cid:1) d ˆ x ρ · d ˆ x σ ⊗ d ˆ x ν + (cid:0) ˜ g ( ρσ )( κλ ) + ˜ g µν Γ µρσ Γ νκλ − ˜ g µ ( κλ ) Γ µρσ − ˜ g ( ρσ ) ν Γ νκλ (cid:1) d ˆ x ρ · d ˆ x σ ⊗ d ˆ x κ · d ˆ x λ = ˆ g µν d ˆ x µ ⊗ d ˆ x ν + ˆ g µ ( κλ ) d ˆ x µ ⊗ d ˆ x κ · d ˆ x λ + ˆ g ( ρσ ) ν d ˆ x ρ · d ˆ x σ ⊗ d ˆ x ν + ˆ g ( ρσ )( κλ ) d ˆ x ρ · d ˆ x σ ⊗ d ˆ x κ · d ˆ x λ = ˆ g ( µρσ ) ( νκλ ) d ˆ x µρσ ⊗ d ˆ x νκλ . (2.42)We notice that a second order vector can be uniquely decomposed in a first order vector and abilinear first order tensor. We will therefore impose˜ g Z = G ◦ g Z ◦ FH ◦ (cid:16) g Z ⊗ g Z (cid:17) ◦ H ∗ ! (2.43)We can then write in a local coordinate systemˆ g ( µρσ ) ( νκλ ) = (cid:18) ˆ g µν ˆ g µ ( κλ ) ˆ g ( ρσ ) ν ˆ g ( ρσ )( κλ ) (cid:19) = (cid:18) g µν ( g ρκ g σλ + g ρλ g σκ ) (cid:19) (2.44)where we have suppressed the maps F , G , H , H ∗ in the second line and where g µν are thecomponents of the first order metric. The inverse can be written asˆ g ( µρσ ) ( νκλ ) = (cid:18) ˆ g µν ˆ g µ ( κλ ) ˆ g ( ρσ ) ν ˆ g ( ρσ )( κλ ) (cid:19) = (cid:18) g µν (cid:0) g ρκ g σλ + g ρλ g σκ (cid:1)(cid:19) (2.45)We can now raise and lower indices on covariant forms and contravariant vectors in the usual14ay ˆ g ( µρσ ) ( νκλ ) ˆ v νκλ = ˆ v µρσ , ˆ g µν ˆ v ν = ˆ v µ , ˆ g ( ρσ )( κλ ) ˆ v κλ = ˆ v ρσ , ˆ g ( µρσ ) ( νκλ ) ˆ ω νκλ = ˆ ω µρσ , ˆ g µν ˆ ω ν = ˆ ω µ , ˆ g ( ρσ )( κλ ) ˆ ω κλ = ˆ ω ρσ , (2.46)where we used the symmetry of v µν and ω µν . Finally we can express the second order metriccomponents ˜ g in terms of the first order metric:˜ g ( µρσ ) ( νκλ ) = (cid:18) ˜ g µν ˜ g µ ( κλ ) ˜ g ( ρσ ) ν ˜ g ( ρσ )( κλ ) (cid:19) = (cid:18) g µν g µα Γ ακλ g αν Γ αρσ ( g ρκ g σλ + g ρλ g σκ ) + g αβ Γ αρσ Γ βκλ (cid:19) (2.47)Its inverse is given by˜ g ( µρσ ) ( νκλ ) = (cid:18) ˜ g µν ˜ g µ ( κλ ) ˜ g ( ρσ ) ν ˜ g ( ρσ )( κλ ) (cid:19) = (cid:18) g µν + g αη g βξ Γ µαβ Γ νηξ − g ακ g βλ Γ µαβ − g ρα g σβ Γ ναβ (cid:0) g ρκ g σλ + g ρλ g σκ (cid:1)(cid:19) (2.48) k -forms In this subsection, we extend the notion of k -forms to the second order geometry framework. Asusual, we denote the bundle of covariant k -tensors by T k ( T ∗ M ) and the subbundle of alternating k -tensors by Λ k ( T ∗ M ). The rank of the latter bundle is (cid:0) nk (cid:1) and a k -form ω ∈ Λ k ( T ∗ M ) can bewritten as ω = ω µ ...µ k dx µ ∧ ... ∧ dx µ k (2.49)where we assume µ < ... < µ k . Similarly, we construct a bundle of second order k -tensors T k ( ˜ T ∗ M ) and a subbundle Λ k ( ˜ T ∗ M ) of rank (cid:0) Nk (cid:1) with N = n ( n + 3). A second order k -formΩ ∈ Λ k ( T ∗ M ) can be written asΩ = ω ( µνρ ) ... ( µνρ ) k d x ( µνρ ) ∧ ... ∧ d x ( µνρ ) k = ω µ ...µ k d x µ ∧ ... ∧ d x µ k + ω ( νρ ) µ ...µ k dx ν · dx ρ ∧ d x µ ∧ ... ∧ d x µ k + ω µ ( νρ ) µ ...µ k d x µ ∧ dx ν · dx ρ ∧ d x µ ∧ ... ∧ d x µ k + ... + ω µ µ ...µ k − ( νρ ) k d x µ ∧ d x µ ∧ ... ∧ d x µ k − ∧ dx ν k · dx ρ k + ω ( νρ ) ( νρ ) µ ...µ k dx ν · dx ρ ∧ dx ν · dx ρ ∧ d x µ ∧ ... ∧ d x µ k + ... + ω ( νρ ) ... ( νρ ) k dx ν · dx ρ ∧ ... ∧ dx ν k · dx ρ k . (2.50)15 .6 Exterior derivatives In this subsection, we extend the notion of the exterior derivative to the second order geometryframework. The first order exterior derivative is a map d : Λ k ( T ∗ M ) → Λ k +1 ( T ∗ M ) such that dω = ∂ ν ω µ ...µ k dx ν ∧ dx µ ∧ ... ∧ dx µ k , (2.51)which is linear: d ( ω + θ ) = dω + dθ ∀ ω, θ ∈ Λ k ( T ∗ M ) ,d ( c ω ) = c dω ∀ ω ∈ Λ k ( T ∗ M ) , c ∈ R ; (2.52)satisfies the modified Leibniz rule: d ( ω ∧ θ ) = dω ∧ θ + ( − k ω ∧ dθ ∀ ω ∈ Λ k ( T ∗ M ) , θ ∈ Λ l ( T ∗ M ); (2.53)satisfies the closure condition d ( d ( ω )) = 0 ∀ ω ∈ Λ k ( T ∗ M ); (2.54)and commutes with pullbacks: φ ∗ ( dω ) = d ( φ ∗ ( ω )) ∀ ω ∈ Λ k ( T ∗ M ) , φ ∈ C ∞ ( M , R ) . (2.55)Analagously we define a a second order exterior derivative d : Λ k ( ˜ T ∗ M ) → Λ k +1 ( ˜ T ∗ M ) suchthat d Ω = ∂ νκλ ω ( µρσ ) ... ( µρσ ) k d x νκλ ∧ d x ( µρσ ) ∧ ... ∧ d x ( µρσ ) k = ∂ ν ω ( µρσ ) ... ( µρσ ) k d x ν ∧ d x ( µρσ ) ∧ ... ∧ d x ( µρσ ) k + ∂ κ ∂ λ ω ( µρσ ) ... ( µρσ ) k dx κ · dx λ ∧ d x ( µρσ ) ∧ ... ∧ d x ( µρσ ) k . (2.56)This second order exterior derivative is also linear and commutes withs pullbacks. Furthermore,it obeys the closure condition d ( d (Ω)) = 0 ∀ Ω ∈ Λ k ( ˜ T ∗ M ); (2.57)and a new modified Leibniz rule d (Ω ∧ Θ) = d Ω ∧ Θ + ( − k Ω ∧ d Θ + 2 d Ω · d Θ ∀ Ω ∈ Λ k ( ˜ T ∗ M ) , Θ ∈ Λ l ( ˜ T ∗ M ) , (2.58)where d Ω · d Θ = ∂ α ω ( µρσ ) ... ( µρσ ) k ∂ β ω ( νκλ ) ... ( νκλ ) l dx α · dx β ∧ d x ( µρσ ) ∧ ... ∧ d x ( µρσ ) k ∧ d x ( νκλ ) ∧ ... ∧ d x ( νκλ ) l . (2.59)The proof for these properties is similar to the proof for the corresponding properties in firstorder geometry, and is therefore omitted. In this subsection, we extend the notion of the interior product to the second order geometryframework. The first order interior product is a map ι v : Λ k ( T ∗ M ) → Λ k − ( T ∗ M ) such that ι v ω = k X l =1 ( − l − v µ l ω µ ...µ k dx µ ∧ ... ∧ dx µ l − ∧ dx µ l +1 ∧ ... ∧ dx µ k . (2.60)16his map is linear, commutes with pullbacks, satisfies the modified Leibniz rule and satisfies theanti-symmetry property { ι u , ι v } ω = 0 . (2.61)Similarly, one can define a second order interior product ι V : Λ k ( T ∗ M ) → Λ k − ( T ∗ M ), suchthat ι V Ω = k X l =1 ( − l − v ( µρσ ) l ω ( µρσ ) ... ( µρσ ) k d x ( µρσ ) ∧ ... ∧ d x ( µρσ ) l − ∧ d x ( µρσ ) l +1 ∧ ... ∧ d x ( µρσ ) k , (2.62)which satisfies the same properties with the modified Leibniz rule replaced by a new modifiedLeibniz rule as in previous subsection. Using the results from previous subsections, we can extend the notion of a Lie derivative to thesecond order geometry framework. A family of diffeomorphisms φ λ := R × M → M satisfyingthe usual (semi-)group properties can be thought of as a vector field v ∈ X ( M ) that generatesa set of integral curves γ v : R → M along the vector field. Along any such integral curveparametrized by λ , one can define the first order derivative of a function f ∈ C ∞ ( M , R ) by ddλ f = dx µ dλ ∂ µ f = v µ ∂ µ f = v f. (2.63)This derivative is equivalent to the Lie derivative along the vector field v L v f = v f, (2.64)which can be generalized to a Lie derivative acting on vectors and forms given by L v u = [ v, u ] , L v ω = { ι v , d } ω. (2.65)In a local coordinate chart, these expressions can be written as L v u µ = v ν ∂ ν u µ − u ν ∂ ν v µ , (2.66) L v ω µ = v ν ∂ ν ω µ + ( ∂ µ v ν ) ω ν . (2.67)Furthermore, using the Leibniz rule one can construct Lie derivatives acting on arbitrary tensorfields.We can analogously define a notion of a Lie derivative of second order tensors along a secondorder vector field V ∈ ˜ X ( M ). As defining relations for derivatives of vectors U ∈ ˜ T x M and formsΩ ∈ ˜ T ∗ x M we take L V f = V f, L V U = [ V, U ] , L V Ω = { ι V , d } Ω . (2.68)In order to make these expressions well defined, we impose v µσ ∂ σ u νρ = u µσ ∂ σ v νρ , (2.69) ω µν = ∂ µ ω ν . (2.70)17n order to satisfy the first condition, we impose u µν = k v µν with k ∈ R and define W ∈ T M ⊂ ˜ T M such that W = kV − U = (cid:18) kv µ − u µ (cid:19) (2.71)In local coordinates we then find L V f = ( v σ ∂ σ + v σκ ∂ σ ∂ κ ) f, L V U µ = ( v σ ∂ σ + v σκ ∂ σ ∂ κ ) u µ − u σ ∂ σ v µ − u σκ ∂ σ ∂ κ v µ , L V U νρ = w σ ∂ σ v νρ − v νσ ∂ σ w ρ − v ρσ ∂ σ w ν , L V Ω µ = ( v σ ∂ σ + v σκ ∂ σ ∂ κ ) ω µ + ω σ ∂ µ v σ + ω σκ ∂ µ v σκ , L V Ω νρ = ( v σ ∂ σ + v σκ ∂ σ ∂ κ ) ω νρ + ω σ ∂ ν ∂ ρ v σ + ω σκ ∂ ν ∂ ρ v σκ + 2 ∂ ( ν v σ ∂ ρ ) ω σ + 2 ∂ ( ν v σκ ∂ ρ ) ω σκ . (2.72)or equivalently with respect to the covariant bases L V f = (ˆ v σ ∇ σ + ˆ v σκ ∇ σ ∇ κ ) f, L V ˆ U µ = (ˆ v σ ∇ σ + ˆ v σκ ∇ σ ∇ κ ) ˆ u µ − ˆ u σ ∇ σ ˆ v µ − ˆ u σκ ∇ σ ∇ κ ˆ v µ + R µσκλ ˆ v σκ ˆ w λ , L V ˆ U νρ = ˆ w σ ∇ σ ˆ v νρ − ˆ v νσ ∇ σ ˆ w ρ − ˆ v ρσ ∇ σ ˆ w ν , L V ˆΩ µ = (ˆ v σ ∇ σ + ˆ v σκ ∇ σ ∇ κ ) ˆ ω µ + ˆ ω σ ∇ µ ˆ v σ + ˆ ω σκ ∇ µ ˆ v σκ + R σκλµ ˆ v κλ ˆ ω σ , L V ˆΩ νρ = (ˆ v σ ∇ σ + ˆ v σκ ∇ σ ∇ κ ) ˆ ω νρ + ˆ ω σ ∇ ( ν ∇ ρ ) ˆ v σ + ˆ ω σκ ∇ ( ν ∇ ρ ) ˆ v σκ + 2 ∇ ( ν | ˆ v σ ∇ | ρ ) ˆ ω σ + 2 ∇ ( ν | ˆ v σκ ∇ | ρ ) ˆ ω σκ − R κ ( νρ ) σ ˆ v σ ˆ ω κ + 2ˆ v σκ (cid:16) R λσκ ( ν ˆ ω ρ ) λ − R λ ( νρ ) σ ˆ ω κλ (cid:17) − ˆ v σκ ˆ ω λ (cid:16) ∇ σ R λ ( νρ ) κ + ∇ ( ν | R λσ | ρ ) κ (cid:17) . (2.73)The Lie derivatives for first order vectors and forms and along first order vector fields can easilybe obtained from these formulae by taking the appropriate limit. Only the Lie derivative of asecond order vector field along a first order vector field cannot be derived as a limit from theseformulae. This one can be obtained by replacing v µν → u µν and w µ → v µ in the above formulae. In this subsection, we discuss the notion of parallel transport along second order vector fields.This notion is similar to the notion of stochastic parallel transport along semi-martingales asdeveloped by Dohrn and Guerra [21, 22]. It is different from first order parallel transport, as thesecond order part of the vector fields generate geodesic deviation. Here, we closely follow thepresentation of stochastic parallel transport by Nelson, cf. section 10 in Ref. [11].Let X ( τ ) be a path in M , passing through the points x, y ∈ M at times τ , τ . We willassume that there exists a convex coordinate chart ( U, χ ) such that x, y ∈ U . Moreover, let V ∈ ˜ T x M be a second order tangent vector at x with ˆ v = F ( V ) its contravariant first orderprojection, such that in χ ( U ) we have y µ = x µ + ˆ v µ .Let d ˆ X ( τ ) ∈ F ( T M ) be a transport and let d ˆ x µ = d ˆ X ( τ ) and d ˆ y µ = d ˆ X ( τ ) be itsvalues when passing through x and y respectively. Then, using the standard notion of paralleltransport, d ˆ X ( τ ) is said to be a parallel transport, if d ˆ y µ = d ˆ x µ − Γ µρσ ( x ) ˆ v ρ d ˆ x σ . (2.74)In order to extend this notion to second order vector fields, we define the difference vector d ˆ v µ := d y µ − d x µ (2.75)18sing the parallel transport equation (2.74), the relations d ˆ x µ = d x µ + Γ µρσ ( x ) d ˆ x ρ · d ˆ x σ ,d ˆ y µ = d y µ + Γ µρσ ( y ) d ˆ y ρ · d ˆ y σ (2.76)and the Taylor expansion Γ µρσ ( y ) = Γ µρσ ( x ) + ∂ ν Γ µρσ ( x )ˆ v ν + O (ˆ v ) , (2.77)we find d ˆ v µ = − Γ µρσ ˆ v ρ d x σ − (cid:0) ∂ ν Γ µρσ + Γ µνκ Γ κρσ − µρκ Γ κνσ (cid:1) ˆ v ν dx ρ · dx σ = − Γ µρσ ˆ v ρ d ˆ x σ − (cid:0) ∂ ν Γ µρσ − µρκ Γ κνσ (cid:1) ˆ v ν d ˆ x ρ · d ˆ x σ (2.78)where Γ µρσ = Γ µρσ ( x ). We will call this the equation of second order parallel transport . Notice thatthe equation of first order parallel transport is obtained if d ˆ X ∈ T M is a first order transportand V ∈ T M is a first order vector, as this implies dx ρ · dx σ = 0 and ˆ v = v respectively.The equation of second order parallel transport is linear in ˆ v and has a solution of the formˆ v µ ( τ ) = P µν ( τ , τ ) ˆ v ν ( τ ) , (2.79)where P µν ( τ , τ ) is the second order parallel propagator . Using this propagator, we can definethe second order directional covariant derivative ˆ d byˆ d ˆ v µ = P µν ( τ , τ ) ˆ v ν ( τ ) − ˆ v µ ( τ )= d ˆ v µ + Γ µρσ ˆ v ρ d ˆ x σ + (cid:0) ∂ ν Γ µρσ − µρκ Γ κνσ (cid:1) ˆ v ν d ˆ x ρ · d ˆ x σ . (2.80) As an aside, we discuss the relation between second order geometry and first order geometryon higher dimensional manifolds. One can embed a n -dimesional pseudo-Riemannian manifoldwith signature ( d, ,
0) into a N -dimensional pseudo-Riemannian manifold ˜ M with signature ( D, n,
0) with N = n ( n + 3) and D = n ( n + 1). We can for example take the trivial embedding ι : M ֒ → ˜ M s . t . ( ι α ( x ) = x α , if α ≤ d ; ι α ( x ) = 0 , if α > d. (2.81)The pushforward ι ∗ of this embedding defines for every x ∈ M a bijection between the secondorder tangent space ˜ T x M and the first order tangent space T ι ( x ) ˜ M . Additionally, the pullback ι ∗ defines a bijection between the cotangent spaces ˜ T ∗ x M and T ∗ ι ( x ) ˜ M . This bijection ι ∗ acts onthe basis vectors as d x µ dx µ ,dx ρ · dx σ dx n + ρ (2 n − ρ − σ . (2.82) We denote the signature by (+,-,0). i.e. ( d, ,
0) corresponds to a ( − + ... +) metric. More generally, if M has signature ( k, l, m ), then ˜ M has signature ( K, L, M ) with K = [ k ( k + 3) + l ( l + 1)], L = l ( k + 1) and M = m (2 k + 2 l + m + 3). Notice that µ ∈ , , ..., d , and that ρ ≤ σ . M and the first ordermetric on ˜ M : ˜ g ( µρσ ) ( νκλ ) ˜ g αβ (2.83)with α, β ∈ { , , ..., N } . One can thus describe the second order geometry framework usingthe first order formalism on a N -dimensional manifold ˜ M instead of the original n -dimensionalmanifold M . However, the support of functions defined on ˜ M must be restricted to the subspace M ⊂ ˜ M . In this section, we discuss stochastic motion on a manifold. Classically, a particle follows atrajectory or path on the manifold, that is parametrized by its proper time. In other words atrajectory is a map γ : T → M , where T = [ τ i , τ f ] ⊂ R .We make this notion stochastic by promoting the manifold to a measurable space ( M , B ( M )),where B ( M ) is the Borel sigma algebra of M . Furthermore, we introduce the probability space(Ω , Σ , P ), and the random variable X : (Ω , Σ , P ) → ( M , B ( M )). Given T = [ τ i , τ f ] ⊂ R we canintroduce a filtration {P τ } τ ∈ T , which is by definition an ordered set such that P τ i ⊆ P s ⊆ P t ⊆ Σ ∀ s < t ∈ T . In addition, we assume the filtration to be right-continuous, i.e. P τ = ∩ ǫ> P τ + ǫ .We can then introduce a stochastic process adapted to this filtration as a family of randomvariables { X ( τ ) : τ ∈ T } . We will restrict the set of stochastic processes to the continuousmanifold valued semi-martingales. These are the continuous manifold valued stochastic processes { X ( τ ) } τ ∈ T such that f ( X ) is a semi-martingle for every smooth function f ∈ C ∞ ( M , R n ). Inparticular, for a coordinate chart χ : U → V with U ⊂ M and V ⊂ R d +1 the coordinates X µ = χ µ ( X ) are semi-martingales. A semi-martingale is a process X ( τ ) that can be decomposedas X ( τ ) = x i + C + ( τ ) + W + ( τ ) , (3.1)where x i := X ( τ i ) is the initial value of the process, C + ( τ ) is a local c`adl`ag process with finitevariation, such that C + ( τ i ) = 0, and W + ( τ ) is a local martingale process, such that W + ( τ i ) = 0,satisfying the martingale property E t + [ W + ( τ )] := E [ W + ( τ ) |{P s } τ i ≤ s ≤ t ] = W + ( t ) ∀ t < τ ∈ T. (3.2)We will make the additional assumption that the time-reversed process is also a semi-martingale. Hence, we can construct a time reversed filtration {F τ } τ ∈ T , which is a left-continuousand decreasing set of sigma algebras, i.e. F τ = ∩ ǫ> F τ − ǫ and F τ f ⊆ F s ⊆ F t ⊆ Σ ∀ s > t ∈ T .Moreover, X is adapted to this filtration and can be decomposed as X ( τ ) = x f + C − ( τ ) + W − ( τ ) , (3.3)where X ( τ f ) = x f , C − ( τ f ) = 0 and W − ( τ f ) = 0. Furthermore, W − satisfies the backwardmartingale property E t − [ W − ( τ )] := E [ W − ( τ ) |{F s } t ≤ s ≤ τ f ] = W − ( t ) ∀ t > τ ∈ T. (3.4)For obvious reasons, we will call {P τ } τ ∈ T the past filtration and {F τ } τ ∈ T the future filtration.The intersection of the two P τ = P τ ∩F τ , will be called the present sigma algebra, and we denoteconditional expectations with respect to this sigma algebra by E t [ X ( τ )] := E [ X ( τ ) | P t ] . (3.5)20urthermore, we will assume Markovianness of both the forward and backward process, i.e. E t + [ X ( τ )] = E t [ X ( τ )] and E t − [ X ( τ )] = E t [ X ( τ )] . (3.6)Finally, one can define a sample path for every ω ∈ Ω as the set γ ( ω ) := { X ( τ, ω ) : τ ∈ T } . The measurable space of sample paths is the cylinder (cid:0) M T , Cyl( M T ) (cid:1) , where we take thecylinder sigma algebra on M T . This construction allows to interpret the stochastic process asa single random variable γ : (Ω , Σ , P ) → (cid:0) M T , Cyl( M T ) (cid:1) , which is essentially the path integralrepresentation that is usually adopted in quantum mechanics. Stochastic motions are not differentiable, and therefore the notion of velocity is not well defined.However, one can define the conditional velocities for the forward and backward process: v µf [ X ( τ ) , τ ] := lim h ↓ h E τ + [ X µ ( τ + h ) − X µ ( τ )] ,v µb [ X ( τ ) , τ ] := lim h ↓ h E τ − [ X µ ( τ − h ) − X µ ( τ )] , (3.7)Using these velocities, we can construct the compensators C ± ( τ ). These c`adl`ag processes aregiven by C µ + ( τ ) = Z ττ i v µf ( X ( s ) , s ) ds,C µ − ( τ ) = Z τ f τ v µb ( X ( s ) , s ) ds. (3.8)Since we are dealing with stochastic processes with non-zero quadratic variation, we can alsodefine v µνf [ X ( τ ) , τ ] := lim h ↓ h E τ + n [ X µ ( τ + h ) − X µ ( τ )][ X ν ( τ + h ) − X ν ( τ )] o ,v µνb [ X ( τ ) , τ ] := lim h ↓ h E τ − n [ X µ ( τ − h ) − X µ ( τ )][ X ν ( τ − h ) − X ν ( τ )] o . (3.9)This can be used to construct the compensator C µν ( τ ) of the quadratic variation process[[ X µ , X ν ]], which is given by C µν + ( τ ) = 2 Z ττ i v µνf ( X ( s ) , s ) ds,C µν − ( τ ) = 2 Z τ f τ v µνb ( X ( s ) , s ) ds. (3.10)In practice, we choose the direction of time. We will therefore introduce a slightly modifiednotion of velocity and define a forward velocity and backward velocity by v + ( X, τ ) = v f ( X, τ ) v − ( X, τ ) = − v b ( X, τ ) (3.11) The compensator of the quadratic variation process is often denoted by the angle bracket h X µ , X ν i . We willuse C µν ( τ ) instead to avoid confusion with the duality pairing. v µ + [ X ( τ ) , τ ] := lim h ↓ h E τ [ X µ ( τ + h ) − X µ ( τ )] ,v µ − [ X ( τ ) , τ ] := lim h ↑ h E τ [ X µ ( τ + h ) − X µ ( τ )] , (3.12)and v µν + [ X ( τ ) , τ ] := lim h ↓ h E τ + n [ X µ ( τ + h ) − X µ ( τ )][ X ν ( τ + h ) − X ν ( τ )] o ,v µν − [ X ( τ ) , τ ] := lim h ↑ h E τ − n [ X µ ( τ + h ) − X µ ( τ )][ X ν ( τ + h ) − X ν ( τ )] o . (3.13)Reversibility of the process imposes v µνb ( τ ) = v µνf ( τ ) , (3.14)and therefore v µν + ( τ ) = − v µν − ( τ ) . (3.15)Moreover, the background hypothesis imposes[[ X µ , X ν ]]( τ ) = ~ m δ νµ τ. (3.16)Hence, d [[ X µ , X ν ]] = ~ m δ νµ dτ. (3.17)Consequently, v µν + [ X ( τ ) , τ ] = 12 dτ E τ h g µρ ( X ( τ )) d [[ X ρ ( τ ) , X ν ( τ )]] i = ~ m g µν ( X ( τ )) . (3.18) v ± [ X ( τ ) , τ ] has the structure of a second order vector, i.e. v ± ( x ) ∈ ˜ T x M . If the metric isfixed , the second order parts v µν ± ( x ) are also fixed. The vectors then live in n -dimensional sub-spaces v µ ± ∈ T ± x M ⊂ ˜ T x M . Since these slices are not invariant under coordinate transformations,we will consider (ˆ v + , ˆ v − ) ∈ ˆ T + x M ⊕ ˆ T − x M instead.Finally, we define a current velocity by v := 12 ( v + + v − ) (3.19)and an osmotic velocity by u := 12 ( v + − v − ) . (3.20)Notice that v ∈ T x M is a first order vector, while u ∈ ˜ T x M has the structure of a second ordervector. Note that the backward velocity can equivalently be defined as v µ − [ X ( τ ) , τ ] := lim h ↓ h E τ [ X µ ( τ ) − X µ ( τ − h )]. In this paper, we only consider test particles in a fixed geometry. .2 Diffeomorphism invariance In classical physics, one imposes a theory to be invariant under diffeomorphisms: general relativ-ity should be invariant under the action of any diffeomorphism φ ∈ C ∞ ( M , N ). The diffeomor-phism φ induces associated maps on the tangent and cotangent spaces, which are the pullback φ ∗ : T ∗ y N → T ∗ x M and the pushforward φ ∗ : T x M → T y N , where y = φ ( x ). The tangent spaceand cotangent space are invariant under respectively the pullback and the pushforward.In quantum physics, we would like to impose the same invariance under diffeomorphisms.However, it is not immediately clear that the n -dimensional tangent subspace ˆ T x M ⊂ ˜ T x M andcotangent subspace ˆ T ∗ x M ⊂ ˜ T ∗ x M with fixed second order parts are invariant spaces under thethe pullback ˜ φ ∗ : ˜ T ∗ y N → ˜ T ∗ x M and pushforward ˜ φ ∗ : ˜ T x M → ˜ T y N of a diffeomorphism φ . Inorder to establish this invariance, we require the notion of a Schwartz morphism: Definition.
Given two manifolds M , N and points x ∈ M , y ∈ N , a linear mapping f : ˜ T x M → ˜ T y N is called a Schwartz morphism , if1. f ( T x M ) ⊂ T y N ,2. ∀ L ∈ ˜ T x M , H ∗ ( f ( L )) = ( f ◦ ⊗ f ◦ ) H ∗ ( L ) ,where f ◦ is the restriction of f to T x M . A Schwartz morphism is thus a morphism that leaves the slices ˆ T x M invariant. Furthermore,it can be shown that a mapping f : ˜ T x M → ˜ T y N is a Schwartz morphism if and only if f = ˜ T x φ for a smooth φ : M → N with φ ( x ) = y . It immediately follows that the pushforward ˜ φ ∗ of adiffeomorphism φ is a Schwartz morphism. Therefore, all slices ˆ T M ⊂ ˜ T M are invariant underthe pushforward ˜ φ ∗ : ˜ T x M → ˜ T φ ( x ) N induced by a diffeomorphism φ : M → N . Moreover, allslices ˆ T ∗ M ⊂ ˜ T ∗ M are invariant under the pullback ˜ φ ∗ : ˜ T ∗ φ ( x ) N → ˜ T ∗ x M of the diffeomorphism φ . We note that this invariance is a consequence of the construction of the ‘covariant slices’ˆ T x M . In the previous sections, we have introduced manifold valued semi-martingales and second ordergeometry. This allows us to construct a notion of integration along semi-martingales on man-ifolds. This section is loosely based on the review by Emery [56]. For mathematical detail werefer to this work by Emery [56] or the original works by Schwartz [54] and Meyer [55].In first order geometry, one defines integrals using forms ω ∈ T ∗ M . The integral of a formalong a curve γ : I → M with I ⊂ R is given by Z γ : T ∗ M → R s . t . ω Z γ ω ( x ) , (4.1)which can be written as Z γ ω = Z τ f τ i ω µ dγ µ = Z τ f τ i ω µ ˙ γ µ dτ, (4.2)where dγ = γ ∗ ( ω ). If we assume that the form can be written as a differential form ω = dF fora function F ∈ C ∞ ( M , R ) we find Z γ dF ( x ) = Z τ f τ i ∂ µ F ( γ ) dγ µ = Z τ f τ i ∂ µ F ( γ ) ˙ γ µ dτ. (4.3) cf. Definition 6.22 in Ref. [56]. cf. Exercise 6.23 in Ref. [56] Z γ dF ( x ) = F [ γ ( τ f )] − F [ γ ( τ i )] . (4.4)One can analogously construct an integral of second order forms Ω ∈ ˜ T ∗ M . The integral ofa second order form along a semi-martingale X can be written as Z X : ˜ T ∗ M → R s . t . Ω Z X Ω( x ) (4.5)with Z X Ω = Z τ f τ i ω µ d X µ + Z τ f τ i ω µν dX µ · dX ν = Z τ f τ i ˆ ω µ d ˆ X µ + Z τ f τ i ˆ ω µν d ˆ X µ · d ˆ X ν , (4.6)where ( d X dXdX ) = X ∗ (Ω). If we assume that the form can be written as a differential formΩ = d F for a function F ∈ C ∞ ( M , R ), we find Z X d F ( x ) = Z τ f τ i ∂ µ F ( X ) d X µ + Z τ f τ i ∂ µ ∂ ν F ( X ) dX µ · dX ν = Z τ f τ i ∇ µ F ( X ) d ˆ X µ + Z τ f τ i ∇ µ ∇ ν F ( X ) d ˆ X µ · d ˆ X ν . (4.7)The fundamental theorem for line integrals can be extended to the second order context, suchthat Z X d F ( x ) = F [ X ( τ f )] − F [ X ( τ i )] , (4.8)Moreover, one can relate the second order integral to first order order integrals. For this weconsider a form ω ∈ T M ⊂ ˜ T M . We can then construct two second order integrals, that aremanifestly invariant under coordinate transformations, using the maps d and G respectively: − Z X ω = Z X dω = Z τ f τ i ω µ d X µ + Z τ f τ i ∂ ν ω µ dX µ · dX ν = Z τ f τ i ˆ ω µ d ˆ X µ + Z τ f τ i ∇ ν ω µ d ˆ X µ · d ˆ X ν , (4.9)and Z X ω = Z X G ( ω )= Z τ f τ i ω µ d X µ + Z τ f τ i ω µ Γ µνρ dX ν · dX ρ = Z τ f τ i ˆ ω µ d ˆ X µ . (4.10) cf. Theorem 6.24 in Ref. [56]. while the second is an Itˆo integral. Weimmediately find a relation between the two − Z X ω = Z X ω + Z τ f τ i ∇ ν ˆ ω µ d ˆ X µ · d ˆ X ν . (4.11)In order to evaluate the integral over the second order part we use that the integral over a bilinearform is given by Z τ f τ i f µν ( X, τ ) dX µ ⊗ dX ν = Z τ f τ i f µν ( X, τ ) d [[ X µ , X ν ]] . (4.12)Using the map H one can then map the integral over the second order part to an integral overa bilinear form. This yields Z τ f τ i f µν ( X, τ ) dX µ · dX ν = 12 Z τ f τ i f µν ( X, τ ) d [[ X µ , X ν ]] = Z τ f τ i f µν ( X, τ ) v µν ( X, τ ) dτ. (4.13)Moreover, if ω can be written as a differential form ω = dF , the two first order integrals canbe written as Z X d F ( x ) = − Z τ f τ i ∂ µ F ( X ) dX µ , Z X d F ( x ) = Z τ f τ i ∇ µ F ( X ) d + ˆ X µ + Z τ f τ i ∇ µ ∇ ν F ( X ) d ˆ X µ · d ˆ X ν . (4.14)Using the decomposition of the semi-martingale, we can then write Z X d F ( x ) = Z τ f τ i v µ ( X, τ ) ∂ µ F ( X ) dτ + − Z τ f τ i ∂ µ F ( X ) dW µ , (4.15) Z X d F ( x ) = Z τ f τ i ˆ v µ + ( X, τ ) ∇ µ F ( X ) dτ + Z τ f τ i ∇ µ F ( X ) dW µ + + Z τ f τ i ˆ v µν + ( X, τ ) ∇ µ ∇ ν F ( X ) dτ. Notice that all integrals are manifestly invariant under coordinate transformations. Furthermore,the Itˆo integral is a local martingale, i.e. E τ + i (cid:20) Z ττ i ∇ µ F ( X ) dW µ + (cid:21) = 0 . (4.16)In addition, we will construct a backward Itˆo integral such that Z X d F ( x ) = Z τ f τ i ∇ µ F ( X ) d − ˆ X µ − Z τ f τ i ∇ µ ∇ ν F ( X ) d ˆ X µ · d ˆ X ν (4.17)= Z τ f τ i ˆ v µ − ( X, τ ) ∇ µ F ( X ) dτ + Z τ f τ i ∇ µ F ( X ) dW µ − + Z τ f τ i ˆ v µν − ( X, τ ) ∇ µ ∇ ν F ( X ) dτ. cf. Definition 7.3 and Proposition 7.4 in Ref. [56]. cf. Definition 7.33 and Proposition 7.34 in Ref. [56]. cf. Theorem 3.8 in Ref. [56]. cf. Proposition 6.31 in Ref. [56]. We use the notation R f µ ( X ) d + ˆ X µ instead of R f µ ( X ) dX µ to make the covariance of the expression explicit. E τ − f (cid:20) Z τ f τ ∇ µ F ( X ) dW µ − (cid:21) = 0 . (4.18)We note that the three integrals are related by − Z X dF ( x ) = 12 (cid:18) Z X dF ( x ) + Z X dF ( x ) (cid:19) . (4.19)Let us now relate the Stratonovich and Itˆo integral to their well known definitions in R n . Ifthere exists a coordinate chart χ : U → R n such that f ([ τ i , τ f ]) ⊂ U , we have − Z τ f τ i f µ ( X, τ ) dX µ := lim k →∞ X [ τ j ,τ j +1 ] ∈ π k h f µ (cid:0) X ( τ j ) , τ j (cid:1) + f µ (cid:0) X ( τ j +1 ) , τ j +1 (cid:1)i × h X µ ( τ j +1 ) − X µ ( τ j ) i , Z τ f τ i f µ ( X, τ ) d + X µ := lim k →∞ X [ τ j ,τ j +1 ] ∈ π k f µ (cid:0) X ( τ j ) , τ j (cid:1)h X µ ( τ j +1 ) − X µ ( τ j ) i , Z τ f τ i f µ ( X, τ ) d − X µ := lim k →∞ X [ τ j ,τ j +1 ] ∈ π k f µ (cid:0) X ( τ j +1 ) , τ j +1 (cid:1)h X µ ( τ j +1 ) − X µ ( τ j ) i , Z τ f τ i f µν ( X, τ ) d [[ X µ , X ν ]] := lim k →∞ X [ τ j ,τ j +1 ] ∈ π k f µν (cid:0) X ( τ j ) , τ j (cid:1)h X µ ( τ j +1 ) − X µ ( τ j ) i × h X ν ( τ j +1 ) − X ν ( τ j ) i , (4.20)where π k is a partition of [ τ i , τ f ], f µ = ( χ ◦ f ) µ and X µ = ( χ ◦ X ) µ . We thus have − Z τ f τ i f µ ( X, τ ) dX µ = 12 (cid:18) Z τ f τ i f µ ( X, τ ) d + X µ + Z τ f τ i f µ ( X, τ ) d − X µ (cid:19) , (4.21)and we will define an osmotic integral by − Z τ f τ i f µ ( X, τ ) d ◦ X µ := 12 (cid:18) Z τ f τ i f µ ( X, τ ) d + X µ − Z τ f τ i f µ ( X, τ ) d − X µ (cid:19) . (4.22) In this subsection, we state two integration by parts formulae, that will be useful for stochasticvariational calculus. The first is given by Z τ f τ i d [ f µ ( τ ) g µ ( τ )] = f µ ( τ f ) g µ ( τ f ) − f µ ( τ i ) g µ ( τ i )= − Z τ f τ i f µ ( τ ) dg µ ( τ ) + − Z τ f τ i g µ ( τ ) df µ ( τ )= Z τ f τ i f µ ( τ ) d + g µ ( τ ) + Z τ f τ i g µ ( τ ) d + f µ ( τ ) + 2 Z τ f τ i df µ ( τ ) · dg µ ( τ )= Z τ f τ i f µ ( τ ) d − g µ ( τ ) + Z τ f τ i g µ ( τ ) d − f µ ( τ ) − Z τ f τ i df µ ( τ ) · dg µ ( τ ) , (4.23) This is a consequence of Theorem 7.14 and Theorem 7.37 in Ref. [56]. f µ ( τ ) = f µ ( X ( τ ) , τ ), g µ ( τ ) = g µ ( X ( τ ) , τ ). We immediately find Z f µ ( τ ) d ◦ g µ ( τ ) + Z g µ ( τ ) d ◦ f µ ( τ ) = − Z df µ ( τ ) · dg µ ( τ ) , (4.24)where we recall Z df µ ( τ ) · dg µ ( τ ) = 12 Z d [[ f µ , g µ ]]( τ ) . (4.25)There exists another integration by parts formula, which can be derived from eq. (4.20) andis given by Z τ f τ i d [ f µ ( τ ) g µ ( τ )] = Z f µ ( τ ) d + g µ ( τ ) + Z g µ ( τ ) d − f µ ( τ )= Z f µ ( τ ) d − g µ ( τ ) + Z g µ ( τ ) d + f µ ( τ ) . (4.26)Combining eqs. (4.24) and (4.26) then yields Z f µ ( τ ) d ◦ g µ ( τ ) = Z g µ ( τ ) d ◦ f µ ( τ ) = − Z df µ ( τ ) · dg µ ( τ ) . (4.27) In this section, we discuss stochastic variational calculus as developed by Yasue [17–19]. We willconsider the tangent bundle ˆ T M = G x ∈M (cid:16) ˆ T + x M ⊕ ˆ T − x M (cid:17) , (5.1)which can be endowed with a (3 n )-dimensional manifold structure with coordinates ( x µ , v µ + , v µ − ).We define the Lagrangian as a map L : ˆ T M → R , (5.2)and the action as the integral S = E (cid:20)Z τ f τ i L ( X, V + , V − ) dτ (cid:21) . (5.3)Equivalently the action can be expressed as a function of the processes X , V ( V + , V − ) and U ( V + , V − ), which we will use later on. We emphasize that V ± ( τ ) are processes on the tangentbundle, while v ± ( X, τ ) are second order vector fields. The two are related as followslim s → τ E τ (cid:2) V µ + ( s ) (cid:3) = v µ + ( X, τ ) , lim s → τ E τ (cid:2) V µ − ( s ) (cid:3) = v µ − ( X, τ ) . (5.4)As we intend to do variational calculus, we require the notion of a norm on the space ofmanifold valued time-reversible semi-martingales. In order to construct such a norm, we wouldlike to split the space of all processes into spaces of time-like, space-like, and null-like processes.For this, we need to define the notion of a time-like process. We will call the process X = X ( τ ) time-like , if g µν ( X ) v µ ( X, τ ) v ν ( X, τ ) < ∀ τ ∈ T. (5.5) See also e.g. Refs. [11, 19] for a derivation of this formula space-like , if g µν ( X ) v µ ( X, τ ) v ν ( X, τ ) > ∀ τ ∈ T, (5.6)and light-like or null-like , if g µν ( X ) v µ ( X, τ ) v ν ( X, τ ) = 0 ∀ τ ∈ T. (5.7)Note that sample paths of a time-like process are not necessarily time-like. Indeed, for atime-like process we have E h g µν ( X ( τ )) dX µ ( τ ) ⊗ dX ν ( τ ) i < ∀ τ ∈ T. (5.8)However, this relation does not hold without the expectation value. Therefore, sample pathscan contain segments that are not time-like. A similar remark holds for space-like and light-likeprocesses.We will now restrict the semi-martingales on M to those that are time-like. After a Wickrotation, the space of these time-like processes can be equipped with the L -norm || X || = s E (cid:20)Z (cid:12)(cid:12) X µ ( τ ) X µ ( τ ) (cid:12)(cid:12) dτ (cid:21) , (5.9)which is the conventional choice in quantum mechanics. The stochastic Euler Lagrange equations can be derived similar to the classical Euler-Lagrangeequations. We vary the action with respect to a semi-martingale δX independent of X thatsatisfies δX ( τ i ) = δX ( τ f ) = 0 . (5.10)This leads to δS ( X ) := S ( X + δX ) − S ( X )= E (cid:20)Z τ f τ i L ( X + δX, V + + δV + , V − + δV − ) dτ (cid:21) − E (cid:20)Z τ f τ i L ( X, V + , V − ) dτ (cid:21) = E (cid:20)Z τ f τ i (cid:26) ∂L ( X, V + , V − ) ∂X µ δX µ + ∂L ( X, V + , V − ) ∂V µ + δV µ + + ∂L ( X, V + , V − ) ∂V µ − δV µ − (cid:27) dτ (cid:21) + O ( || δX || )= E (cid:20)Z τ f τ i (cid:26) ∂L ( X, V + , V − ) ∂X µ δX µ dτ + ∂L ( X, V + , V − ) ∂V µ + d + δX µ + ∂L ( X, V + , V − ) ∂V µ − d − δX µ (cid:27)(cid:21) + O ( || δX || )= E (cid:20)Z τ f τ i δX µ (cid:26) ∂L ( X, V + , V − ) ∂X µ dτ − d − ∂L ( X, V + , V − ) ∂V µ + − d + ∂L ( X, V + , V − ) ∂V µ − (cid:27)(cid:21) + O ( || δX || ) , (5.11)28here we used the partial integration formula (4.26). We find a system of stochastic differentialequations given by Z τ f τ i ∂∂X µ L ( X, V + , V − ) dτ = Z τ f τ i (cid:26) d − ∂∂V µ + L ( X, V + , V − ) + d + ∂∂V µ − L ( X, V + , V − ) (cid:27) (5.12)or equivalently Z τ f τ i ∂∂X µ L ( X, V, U ) dτ = Z τ f τ i (cid:26) d ∂∂V µ L ( X, V, U ) − d ◦ ∂∂U µ L ( X, V, U ) (cid:27) . (5.13)Since δX ⊥⊥ X , the osmotic integral vanishes, and we obtain Z τ f τ i ∂∂X µ L ( X, V, U ) dτ = Z τ f τ i d ∂∂V µ L ( X, V, U ) . (5.14) As in classical physics, one can define an Hamiltonian picture. We define the generalized momentaby P + µ ( τ ) = ∂L∂V µ + ,P − µ ( τ ) = ∂L∂V µ − . (5.15)and the Hamiltonian as the Legendre transform H ( X, P + , P − ) = P + µ V µ + + P − µ V µ − − L ( X, V + , V − ) . (5.16)We can take a first order total derivative. This yields dH = ∂H∂X µ dX µ + ∂H∂P + µ dP + µ + ∂H∂P − µ dP − µ (5.17)and dH = P + µ dV µ + + V µ + dP + µ + P − µ dV µ − + V µ − dP − µ − ∂L∂X µ dX µ − ∂L∂V µ + dV µ + − ∂L∂V µ − dV µ − = V µ + dP + µ + V µ − dP − µ − (cid:18) d − dτ P + µ + d + dτ P − µ (cid:19) dX µ . (5.18)One can then read off the Hamilton equations: V µ + ( τ ) = ∂H∂P + µ ,V µ − ( τ ) = ∂H∂P − µ (5.19)and Z (cid:0) d + P − µ + d − P + µ (cid:1) = − Z ∂H∂X µ dτ. (5.20)29urthermore, if an explicit proper time dependence is introduced, one finds ∂∂τ H ( X, P + , P − , τ ) = − ∂∂τ L ( X, V + , V − , τ ) . (5.21)As is the case for the Lagrangian, one can express the Hamiltonian in terms of current and osmotic momenta . These can be defined as P µ ( τ ) = ∂∂V µ L ( X, V, U ) ,Q µ ( τ ) = ∂∂U µ L ( X, V, U ) . (5.22)The Hamiltonian is then given by H ( X, P, Q ) = P µ V µ + Q µ U µ − L ( X, V, U ) . (5.23)This leads to the Hamilton equations V µ ( τ ) = ∂H∂P µ ,U µ ( τ ) = ∂H∂Q µ . (5.24)and Z dP µ = − Z ∂H∂X µ dτ. (5.25)Let us summarize the relation between U, V, V + , V − : V = 12 ( V + + V − ) , V + = V + U,U = 12 ( V + − V − ) , V − = V − U. (5.26)Furthermore, for P, Q, P + , P − we have P = P + + P − , P + = 12 ( P + Q ) ,Q = P + − P − , P − = 12 ( P − Q ) . (5.27) The Hamilton-Jacobi equations play an important role in the derivation of the Schr¨odingerequation in stochastic quantization. We will therefore review the derivation of these equations.We define Hamilton’s principal function as the action conditioned on its end point S ( X, τ ) = E (cid:20)Z ττ i L ( X, V + , V − ) dt (cid:12)(cid:12)(cid:12) X ( τ ) (cid:21) , (5.28)such that the Euler-Lagrange equations are satisfied.30e consider the variation of the principal function under a variation of the end point. Thisyields δS ( X, τ ) = S ( X + δX, τ ) − S ( X, τ )= E (cid:20)Z ττ i L ( X, V + , V − ) dt (cid:12)(cid:12)(cid:12) X ( τ ) + δX ( τ ) (cid:21) − E (cid:20)Z ττ i L ( X, V + , V − ) dt (cid:12)(cid:12)(cid:12) X ( τ ) (cid:21) = E (cid:20)Z ττ i L ( X + δX, V + + δV + , V − + δV − ) dt − Z ττ i L ( X, V + , V − ) dt (cid:12)(cid:12)(cid:12) X ( τ ) , δX ( τ ) (cid:21) = E (cid:20)Z ττ i (cid:26) ∂∂X µ L ( X, V + , V − ) δX µ + ∂∂V µ + L ( X, V + , V − ) δV µ + + ∂∂V µ − L ( X, V + , V − ) δV µ − (cid:27) dt (cid:12)(cid:12)(cid:12) X ( τ ) , δX ( τ ) (cid:21) + O (cid:0) || δX || (cid:1) = E (cid:20)Z ττ i (cid:26) δX µ d − ∂L∂V µ + + δX µ d + ∂L∂V µ − + ∂L∂V µ + d + δX µ + ∂L∂V µ − d − δX µ (cid:27) (cid:12)(cid:12)(cid:12) X ( τ ) , δX ( τ ) (cid:21) + O (cid:0) || δX || (cid:1) = E (cid:20)Z ττ i d (cid:20)(cid:18) ∂L∂V µ + + ∂L∂V µ − (cid:19) δX µ (cid:21) + O (cid:0) || δX || (cid:1) (cid:12)(cid:12)(cid:12) X ( τ ) , δX ( τ ) (cid:21) = (cid:16) p + µ ( X, τ ) + p − µ ( X, τ ) (cid:17) δX µ + O (cid:16) || δX || (cid:17) , (5.29)where we used the Euler-Lagrange equations in the fifth line. Furthermore, in the third line, wehave rewritten the original trajectory which is the minimal path between ( τ i , x i ) and ( τ, X ( τ ) + δX ( τ )) as two independent trajectories X, δX , which are the minimal paths between ( τ i , x i ) and( τ, X ( τ )) and between ( τ i ,
0) and ( τ, δX ( τ )) respectively.We conclude with the first Hamilton-Jacobi equation ∇ µ S ( X, τ ) = p + µ ( X, τ ) + p − µ ( X, τ ) = p µ ( X, τ ) . (5.30)Moreover, taking a first order total derivative of Hamilton’s principal function yields dS = E τ [ L dτ ] ,dS = E τ (cid:20) ∂S∂x µ dX µ + ∂S∂τ dτ (cid:21) . (5.31)This leads to the second Hamilton-Jacobi equation ∂∂τ S ( X, τ ) = E τ [ L ( X, V, U )] − p µ v µ . (5.32) In this section, we derive the Kolmogorov equations. Although these do not follow from avariational principle, they are another crucial ingredient for the derivation of the Schr¨odingerequation.Let µ ( x, τ ) be a probability measure on M × T , such that Z M× T f ( x, τ ) dµ ( x, τ ) = Z T E [ f ( X ( τ ) , τ )] dτ (5.33)31or any smooth function f compactly supported on M × int(T), where int(T) is the interior of T . We will assume that the probability density ρ associated to the measure µ exists, such that dµ ( x, τ ) = p | g | ρ ( x, τ ) d n xdτ . Then0 = E [ f ( X ( τ f ) , τ f )] − E [[ f ( X ( τ i ) , τ i )]= Z T d dτ E [ f ( X ( τ ) , τ )] dτ = Z T E (cid:20) d dτ f ( X ( τ ) , τ ) (cid:21) dτ = Z T E (cid:20) E τ (cid:20) d dτ f ( X ( τ ) , τ ) (cid:21)(cid:21) dτ = Z T E (cid:20)(cid:18) ∂∂τ + ˆ v µ ( X, τ ) ∇ µ + ˆ v µν ( X, τ ) ∇ µ ∇ ν (cid:19) f ( X, t ) (cid:21) dτ = Z M× T (cid:18) ∂∂τ + ˆ v µ ( x, τ ) ∇ µ + ˆ v µν ( x, τ ) ∇ µ ∇ ν (cid:19) f ( x, τ ) dµ ( x, τ )= Z M× T p | g | ρ ( x, τ ) (cid:18) ∂∂τ + ˆ v µ ( x, τ ) ∇ µ + ˆ v µν ( x, τ ) ∇ µ ∇ ν (cid:19) f ( x, τ ) d n x dτ = Z M× T p | g | f ( x, τ ) (cid:18) − ∂∂τ − ˆ v µ ( x, τ ) ∇ µ + ˆ v µν ( x, τ ) ∇ µ ∇ ν (cid:19) ρ ( x, τ ) d n x dτ (5.34)for all compactly supported functions f . We can choose v = v ± , and plug in the backgroundhypothesis ˆ v µν ± = ± ~ m g µν . (5.35)This leads to the Kolmogorov forward and backward equations or equivalently the Fokker-Planckequations associated to the forward and backward process: ∂∂τ ρ ( x, τ ) = −∇ µ (cid:2) ˆ v µ + ( x, τ ) ρ ( x, τ ) (cid:3) + ~ m ∇ ρ ( x, τ ) ,∂∂τ ρ ( x, τ ) = −∇ µ (cid:2) ˆ v µ − ( x, τ ) ρ ( x, τ ) (cid:3) − ~ m ∇ ρ ( x, τ ) . (5.36)Adding and subtracting the two equations leads to the continuity and osmotic equations ∂∂τ ρ ( x, τ ) = −∇ µ [ v µ ( x, τ ) ρ ( x, τ )] , (5.37)ˆ u µ ( x, τ ) = ~ m ∇ µ ln [ ρ ( x, τ )] . (5.38) In classical physics a Lagrangian is a function of the form L ( X, V, τ ). In stochastic quantizationon the other hand the Lagrangian is a function of the form L ( X, V + , V − , τ ). Due to the existenceof two different velocities, it is not immediately clear how the classical Lagrangian should begeneralized to the stochastic framework. However, it was shown by Zambrini, cf. Ref. [19] thatfor any classical Lagrangian of the form L c ( x, v, τ ) = m T µν ( x, τ ) v µ v ν − ~ A µ ( x, τ ) v µ − U ( x, τ ) (6.1)32he minimal stochastic extension that is compatible with gauge invariance and Maupertuis’principle is given by L ( X, V + , V − , τ ) = 12 L c ( X, V + , τ ) + 12 L c ( X, V − , τ ) . (6.2)We note that this form of the Lagrangian was also assumed by Yasue [17, 18]. In the remainderof this paper, we will assume that gravity is the only spin-2 field, i.e. T µν ( x, τ ) = g µν ( x ) . (6.3)The stochastic Lagrangian corresponding to the classical Lagrangian (6.1) is then given by L ( X, V + , V − ) = m g µν (cid:0) V µ + V ν + + V µ − V ν − (cid:1) − ~ A µ ( X ) (cid:0) V µ + + V µ − (cid:1) − U ( X ) (6.4)or equivalently L ( X, V, U ) = m g µν ( V µ V ν + U µ U ν ) − ~ A µ ( X ) V µ − U ( X ) . (6.5)Compared to the classical Lagrangian there is an additional energy contribution: m g µν U µ U ν . (6.6)This is the osmotic energy and can be interpreted as the kinetic energy of the background field.There also exists a Hamiltonian description. The momenta for this Lagrangian are P + µ ( τ ) = m g µν V ν + ( τ ) − ~ A µ ( X ) ,P − µ ( τ ) = m g µν V ν − ( τ ) − ~ A µ ( X ) ,P µ ( τ ) = m g µν V ν ( τ ) − ~ A µ ( X ) ,Q µ ( τ ) = m g µν U ν ( τ ) . (6.7)The Hamiltonian is then given by H (cid:0) X, P + , P − (cid:1) = 1 m g µν (cid:18) P + µ P + ν + P − µ P − ν + ~ (cid:0) P + µ + P − µ (cid:1) A ν ( X ) + ~ A µ ( X ) A ν ( X ) (cid:19) + U ( X )(6.8)or equivalently H ( X, P, Q ) = 12 m g µν (cid:0) P µ P ν + Q µ Q ν + 2 ~ P µ A ν ( X ) + ~ A µ ( X ) A ν ( X ) (cid:1) + U ( X ) . (6.9) In section 5.3, we derived the Hamilton-Jacobi equations and obtained expressions that containedthe conditional expectation of the Lagrangian E τ [ L ( X, V, U, τ )]. We can calculate this expressionfor the Lagrangian (6.4) obtained in previous subsection. For this we notice that for any smoothfunction U : T × M → RE τ [ U ( X ( τ ) , τ )] = lim s → τ E τ [ U ( X ( s ) , s )] = U ( X ( τ ) , τ ) . (6.10)33or the terms that depend on the velocity process, we need to make sense of the processes V ± .This can be done by performing an integration over dτ . At linear order we have A µ ( X ( τ ) , τ ) V µ + ( τ ) = lim h → h Z τ + hτ A µ ( X ( s ) , s ) V µ + ( s ) ds = lim h → h " Z τ + hτ ( A µ ( X ( s ) , s ) d + X µ ( s ) + ∂ ν A µ ( X ( s ) , s ) dX µ · dX ν ( s )) = lim h → h " Z τ + hτ (cid:16) A µ ( X ( s ) , s ) d + ˆ X µ ( s ) + ∇ ν A µ ( X ( s ) , s ) d ˆ X µ · d ˆ X ν ( s ) (cid:17) . (6.11)By a similar calculation, we obtain A µ ( X ( τ ) , τ ) V µ − ( τ ) = lim h → h " Z τ + hτ (cid:16) A µ ( X ( s ) , s ) d − ˆ X µ ( s ) − ∇ ν A µ ( X ( s ) , s ) d ˆ X µ · d ˆ X ν ( s ) (cid:17) . (6.12)We note that we can write these expressions in differential notation as A µ V µ ± dτ = A µ d ± ˆ X µ ± ∇ ν A µ d ˆ X ν · d ˆ X ν (6.13)Taking the expectation value of these expressions yields E τ (cid:2) A µ ( X ( τ ) , τ ) V µ + ( τ ) (cid:3) = lim h → h E τ "Z τ + hτ A µ ( X ( s ) , s ) ˆ v µ + ( X ( s ) , s ) ds + Z τ + hτ A µ ( X ( s ) , s ) dW µ + ( s )+ Z τ + hτ ∇ ν A µ ( X ( s ) , s ) ˆ v µν + ( X ( s ) , s ) ds = A µ ( X ( τ ) , τ ) ˆ v µ + ( X, τ ) + ~ m ∇ µ A µ ( X ( τ ) , τ ) , (6.14)where we used the martingale property (4.16). Moreover, E τ (cid:2) A µ ( X ( τ ) , τ ) V µ − ( τ ) (cid:3) = A µ ( X ( τ ) , τ ) ˆ v µ − ( X, τ ) − ~ m ∇ µ A µ ( X ( τ ) , τ ) . (6.15)Consequently, E τ [ A µ ( X ( τ ) , τ ) V µ ( τ )] = A µ ( X ( τ ) , τ ) v µ ( X, τ ) , (6.16) E τ [ A µ ( X ( τ ) , τ ) U µ ( τ )] = A µ ( X ( τ ) , τ ) ˆ u µ ( X, τ ) + ~ m ∇ µ A µ ( X ( τ ) , τ ) . (6.17)For the terms quadratic in velocity we will perform a double integral over dτ . In differential34otation we have g µν V µ + V ν + dτ = g µν d + ˆ X µ ⊗ d + ˆ X ν + g µν ∇ ρ (cid:16) d + ˆ X µ (cid:17) ⊗ d ˆ X ν · d ˆ X ρ + g µν d ˆ X µ · d ˆ X ρ ⊗ ∇ ρ (cid:16) d + ˆ X ν (cid:17) − R µνρσ d ˆ X µ · d ˆ X ρ ⊗ d ˆ X ν · d ˆ X σ ,g µν V µ − V ν − dτ = g µν d − ˆ X µ ⊗ d − ˆ X ν − g µν ∇ ρ (cid:16) d − ˆ X µ (cid:17) ⊗ d ˆ X ν · d ˆ X ρ − g µν d ˆ X µ · d ˆ X ρ ⊗ ∇ ρ (cid:16) d − ˆ X ν (cid:17) − R µνρσ d ˆ X µ · d ˆ X ρ ⊗ d ˆ X ν · d ˆ X σ ,g µν V µ + V ν − dτ = g µν d + ˆ X µ ⊗ d − ˆ X ν − g µν ∇ ρ (cid:16) d + ˆ X µ (cid:17) ⊗ d ˆ X ν · d ˆ X ρ + g µν d ˆ X µ · d ˆ X ρ ⊗ ∇ ρ (cid:16) d − ˆ X ν (cid:17) + 23 R µνρσ d ˆ X µ · d ˆ X ρ ⊗ d ˆ X ν · d ˆ X σ . (6.18)We can take the expectation values of these expressions. This yields E τ h g µν d + ˆ X µ ⊗ d + ˆ X ν i = E τ h g µν (cid:16) ˆ v µ + ˆ v ν + dτ + ˆ v µ + dW ν + dτ + ˆ v ν + dW ν + dτ + dW µ + ⊗ dW ν + (cid:17)i = g µν (cid:16) ˆ v µ + ˆ v ν + dτ + 2 ˆ v µν + dτ (cid:17) = n ~ m dτ + g µν ˆ v µ + ˆ v ν + dτ (6.19)where we used that the expectation value of the terms linear in dW + vanishes, due to themartingale property of W + . Moreover, we used eq. (4.12) to evaluate the term dW µ + dW ν + = dW µ + ⊗ dW ν + . By a similar calculation we obtain E τ h g µν d − ˆ X µ ⊗ d − ˆ X ν i = − n ~ m dτ + g µν ˆ v µ − ˆ v ν − dτ (6.20) E τ h g µν d + ˆ X µ ⊗ d − ˆ X ν i = g µν ˆ v µ + ˆ v ν − dτ . (6.21)Furthermore, E τ h g µν ∇ ρ (cid:16) d + ˆ X µ (cid:17) ⊗ d ˆ X ν · d ˆ X ρ i = E τ h g µν ˆ v νρ + ∇ ρ (cid:0) ˆ v µ + dτ + dW µ + (cid:1) dτ i = ~ m ∇ µ ˆ v µ + dτ . (6.22)Similarly, E τ h g µν ∇ ρ (cid:16) d − ˆ X µ (cid:17) ⊗ d ˆ X ν · d ˆ X ρ i = ~ m ∇ µ ˆ v µ − dτ . (6.23)For the remaining term we find E τ h R µνρσ d ˆ X µ · d ˆ X ρ ⊗ d ˆ X ν · d ˆ X σ i = E τ (cid:2) R µνρσ ˆ v µρ ˆ v νσ dτ (cid:3) = ~ m R dτ . (6.24) cf. section 9 in Ref. [11].
35e conclude, E τ (cid:2) g µν V µ + V ν + (cid:3) = g µν ˆ v µ + ˆ v ν + + ~ m ∇ µ ˆ v µ + − ~ m R + n ~ m dτ , E τ (cid:2) g µν V µ − V ν − (cid:3) = g µν ˆ v µ − ˆ v ν − − ~ m ∇ µ ˆ v µ − − ~ m R − n ~ m dτ , E τ (cid:2) g µν V µ + V ν − (cid:3) = g µν ˆ v µ + ˆ v ν − − ~ m ∇ µ ˆ v µ + + ~ m ∇ µ ˆ v µ − + ~ m R (6.25)or equivalently E τ h g µν V µ V ν i = g µν v µ v ν , E τ [ g µν U µ U ν ] = g µν ˆ u µ ˆ u ν + ~ m ∇ µ ˆ u µ − ~ m R , E τ [ g µν V µ U ν ] = g µν v µ ˆ u ν + ~ m ∇ µ v µ + n ~ m dτ . (6.26)The conditional expectation of the Lagrangian (6.5) is thus given by E τ [ L ( X, V, U, τ )] = m g µν ( v µ v ν + ˆ u µ ˆ u ν ) + ~ ∇ µ ˆ u µ − ~ m R − ~ A µ v µ − U . (6.27) Observables in quantum mechanics can be constructed from correlation functions computed in thepath integral formalism. Since this computation is slightly different in stochastic quantization,we review the main steps.In order to compute correlation functions within the stochastic quantization, one must firstsolve the stochastic equations of motion derived from the action. The solution is a stochasticprocess { X ( τ ) | τ ∈ T } . For this stochastic process one can define a characteristic functionalΦ X ( J ), and a moment generating functional M X ( J ):Φ X ( J ) = E h e i ~ R τfτi J µ ( τ ) X µ ( τ ) dτ i , (6.28) M X ( J ) = E h e ~ R τfτi J µ ( τ ) X µ ( τ ) dτ i , (6.29)where J ( τ ) is a bounded process of finite variation that corresponds to the source in the pathintegral formulation. We emphasize that one no longer averages over the action, as this isessentially done in the first step, where the stochastic differential equation is solved.Using the characteristic and moment generating functionals for the process X ( τ ), one cancalculate all moments of the theory. For example, the two-point correlation function is given by E [ X µ ( s ) X ν ( t )] = lim || J ||→ ∂∂J µ ( s ) ∂∂J ν ( t ) M X ( J ) . (6.30)We emphasize that the integrals that need to be evaluated in the path integral formalismand stochastic quantization are constructed in different ways. Due to this different construction,theories that require renormalization in the path integral formalism can be finite in stochasticquantization. Note that the divergent term n ~ mdτ does not appear in the Lagrangian. .3 Uncertainty principle Due to the relevance of the unvertainty principle in quantum mechanics, we will derive it instochastic quantization, which can be done using the results from section 6.1.For s > τ we findCov τ [ X µ ( s ) , X ν ( s )] = E τ [ X µ ( s ) X ν ( s )] − E τ [ X µ ( s )] E τ [ X ν ( s )]= E τ (cid:20)(cid:18) X µ ( τ ) + Z sτ V + µ ( r ) dr (cid:19) (cid:18) X µ ( τ ) + Z sτ V µ + ( r ) dr (cid:19)(cid:21) − E τ (cid:20)(cid:18) X µ ( τ ) + Z sτ V µ ( r ) dr (cid:19)(cid:21) E τ (cid:20)(cid:18) X µ ( τ ) + Z sτ V µ ( r ) dr (cid:19)(cid:21) = ~ m δ νµ ( s − τ ) + ~ m (cid:18) ∇ µ ˆ v ν + + ∇ ν ˆ v + µ − ~ m R νµ (cid:19) ( s − τ ) + o ( s − τ ) . (6.31)Furthermore, the covariance for the momenta is given byCov τ (cid:2) P + µ ( s ) , P + ν ( s ) (cid:3) = m n E τ (cid:2) V + µ ( s ) V ν + ( s ) (cid:3) − E τ [ V + µ ( s )] E τ (cid:2) V ν + ( s ) (cid:3) o − m ~ n E τ [ V + µ ( s ) A ν ( s )] − E τ [ V + µ ( s )] E τ [ A ν ( s )] o − m ~ n E τ (cid:2) A µ ( s ) V ν + ( s ) (cid:3) − E τ [ A µ ( s )] E τ (cid:2) V ν + ( s ) (cid:3) o − ~ n E τ [ A µ ( s ) A ν ( s )] − E τ [ A µ ( s )] E τ [ A ν ( s )] o = m ~ δ νµ ( s − τ ) − + m ~ (cid:0) ∇ µ ˆ v ν + + ∇ ν ˆ v + µ (cid:1) − ~ ∇ µ A ν + ∇ ν A µ ) − ~ R νµ + o (1) . (6.32)If we take the limit s → τ , we findlim s → τ Cov τ [ X µ ( s ) , X ν ( s )] = 0 , (6.33)lim s → τ Cov τ (cid:2) P + µ ( s ) , P + ν ( s ) (cid:3) = ∞ . (6.34)This reflects the fact that we have constructed the stochastic theory in a position representation,i.e. the process ( X, P + , P − ) is adapted to the filtration generated by the process X .We can calculate the product of the two variances. For this we fix the indices µ = ν = ¯ µ ,and obtainVar τ (cid:2) X ¯ µ ( s ) (cid:3) Var τ (cid:2) P +¯ µ ( s ) (cid:3) = ~ ~ (cid:18) ∇ ¯ µ ˆ v ¯ µ + − ~ m ∇ ¯ µ A ¯ µ − ~ m R ¯ µ ¯ µ (cid:19) ( s − τ ) + o ( s − τ ) . (6.35)If we then take the limit s → τ , we findlim s → τ Var τ (cid:2) X ¯ µ ( s ) (cid:3) Var τ (cid:2) P +¯ µ ( s ) (cid:3) = ~ . (6.36)This corresponds to the lower bound given by the Heisenberg uncertainty principle.37 Scalar Test Particles
In this section, we derive the equations of motion that govern a quantum mechanical spin-0 testparticle on a pseudo-Riemannian manifold subjected to the Lagrangian (6.5).
We consider the Lagrangian (6.5): L ( X, V, U ) = m g µν ( V µ V ν + U µ U ν ) − ~ A µ V µ − U . (7.1)After integrating this expression twice over τ we obtain, cf. eq. (6.18), E (cid:2) L dτ (cid:3) = E h m g µν n dX µ dX ν + d ◦ ˆ X µ d ◦ ˆ X ν + ∇ ρ (cid:16) d ◦ ˆ X µ (cid:17) d [[ X ν , X ρ ]] − ~ m R µρκσ d [[ X ν , X κ ]] d [[ X ρ , X σ ]] (cid:27) − ~ A µ dX µ dτ − U dτ (cid:3) = E (cid:20) m g µν dX µ dX ν − ~ A µ dX µ dτ − (cid:18) U + ~ m R (cid:19) dτ (cid:21) , (7.2)where we used E h g µν n d ◦ ˆ X µ d ◦ ˆ X ν + ∇ ρ (cid:16) d ◦ ˆ X µ (cid:17) d [[ X ν , X ρ ]] oi = 0 , (7.3)which follows from eq. (4.27) and the metric compatibility. If we vary this expression with respectto a stochastically independent deviation process δX , we obtain the stochastic Euler-Lagrangeequations (5.14) that take the form m (cid:0) g µν d X ν + g µν Γ νρσ dX ρ dX σ (cid:1) = (cid:18) ~ ∂ τ A µ − ∇ µ U − ~ m ∇ µ R (cid:19) dτ − ~ H µν dX ν dτ, (7.4)where H µν := ∂ µ A ν − ∂ ν A µ = ∇ µ A ν − ∇ ν A µ . (7.5)In the classical limit ~ →
0, the quadratic variation vanishes. This gives m (cid:18) g µν d X ν dτ + g µν Γ νρσ dX ρ dτ dX σ dτ (cid:19) = − lim ~ → (cid:26) ∇ µ U + ~ (cid:20) − ∂ τ A µ + H µν dX ν dτ (cid:21)(cid:27) , (7.6)which is consistent with general relativity. On the other hand, taking the flat space-time limit G N → g µν = η µν , and therefore m η µν d X ν = (cid:0) ~ ∂ τ A µ − ∂ µ U (cid:1) dτ − ~ H µν dX ν dτ. (7.7)If we then take the non-relativistic limit c → ∞ , we identify t = τ and replace η µν → δ ij . Theresulting equation is consistent with stochastic quantization in flat spaces [8–11, 19, 28].The stochastic differential equation (7.4) is the fundamental equation of motion in stochasticquantization. The solutions describe the stochastic trajectories of quantum mechanical spin-0test particles in any geometry. In section 7.3, we will show that probability density functionassociated to the solution X ( τ ) of this equation evolves according to the Schr¨odinger equation. Note that U and A µ could contain an additional ~ dependence. .2 Stochastic Newton equation The stochastic differential equation derived in previous section can be rewritten as a diffusionequation for the vector fields v ± ( x, τ ). This representation is known as the stochastic Newtonequation, see e.g. Ref. [11]. In order to derive it, we define a function R ( x, τ ) := ~ ρ ( x, τ )] . (7.8)The osmotic (5.38) and continuity equation (5.37) can then be rewritten as ∇ µ R ( x, τ ) = m ˆ u µ , (7.9) ∂∂τ R ( x, τ ) = − (cid:18) m g µν ˆ u ν + ~ ∇ µ (cid:19) ˆ v µ . (7.10)Furthermore, we recall that the Hamilton Jacobi equations (5.30) and (5.32) are given by ∇ µ S ( x, τ ) = p µ , (7.11) ∂∂τ S ( x, τ ) = E τ [ L ( X, V, U, τ )] − p µ v µ . (7.12)We consider the Lagrangian (6.5) L ( X, V, U, τ ) = m g µν ( V µ V µ + U µ U µ ) − ~ A µ V µ − U (7.13)with momenta P µ ( τ ) = m g µν V ν − ~ A µ ,Q µ ( τ ) = m g µν U ν . (7.14)Therefore, p µ ( x, τ ) = E τ [ P µ ( τ )] = m g µν v ν − ~ A µ , ˆ q µ ( x, τ ) = E τ [ Q µ ( τ )] = m g µν ˆ u ν . (7.15)Moreover, in eq. (6.27), we found E τ [ L ( X, V, U, τ )] = m g µν ( v µ v ν + ˆ u µ ˆ u ν ) + ~ ∇ µ ˆ u µ − ~ m R − ~ A µ v µ − U . (7.16)Putting everything together yields ∇ µ S ( x, τ ) = p µ = m g µν v ν − ~ A µ , (7.17) ∇ µ R ( x, τ ) = ˆ q µ = m g µν ˆ u ν (7.18)and ∂∂τ S ( x, τ ) = − m g µν ( v µ v ν − ˆ u µ ˆ u ν ) + ~ ∇ µ ˆ u µ − ~ m R − U , (7.19) ∂∂τ R ( x, τ ) = − m g µν v µ ˆ u µ − ~ ∇ µ v µ . (7.20)39e take the covariant derivative of eq. (7.19). This yields m ∂v µ ∂τ − ~ ∂A µ ∂τ = − m v ν ∇ µ v ν + m ˆ u ν ∇ µ ˆ u ν + ~ ∇ µ ∇ ν ˆ u ν − ~ m ∇ µ R − ∇ µ U . (7.21)Using eqs. (7.17) and (7.18), we find ∇ µ ˆ u ν = ∇ ν ˆ u µ , ∇ µ v ν = ∇ ν v µ + ~ m H µν , ∇ µ ∇ ν ˆ u ν = (cid:3) ˆ u µ − R µν ˆ u ν . (7.22)Therefore, ~ (cid:18) ∂A µ ∂τ − H µν ˆ v ν (cid:19) − ~ m ∇ µ R − ∇ µ U = m (cid:18) ∂ ˆ v µ ∂τ + v ν ∇ ν ˆ v µ − ˆ u ν ∇ ν ˆ u µ − ˆ u ρσ ∇ ρ ∇ σ ˆ u µ + ˆ u ρσ R νρσµ ˆ u ν (cid:17) . (7.23)We will associate the left hand side with a force, i.e. F µ := ~ (cid:18) ∂A µ ∂τ − H µν ˆ v ν (cid:19) − ~ m ∇ µ R − ∇ µ U . (7.24)Moreover, we rewrite the left hand side in terms of the forward and backward velocity. We find F µ = m (cid:20)(cid:18) ∂∂τ + ˆ v ν + ∇ ν + ˆ v ρσ + ∇ ρ ∇ σ (cid:19) ˆ v µ − − R µρσν ˆ v ρσ + ˆ v ν − + (cid:18) ∂∂τ + ˆ v ν − ∇ ν + ˆ v νρ − ∇ ν ∇ ρ (cid:19) ˆ v µ + − R µρσν ˆ v ρσ − ˆ v ν + (cid:21) . (7.25)As we would like to associate the right hand side with an acceleration, we define second orderacceleration vectors a ±± by a µ + ± ( x, τ ) := lim h → h E τ (cid:2) V µ ± ( τ + h ) − V µ ± ( τ ) (cid:3) ,a µ −± ( x, τ ) := lim h → h E τ (cid:2) V µ ± ( τ ) − V µ ± ( τ − h ) (cid:3) , (7.26)and a ρσ + ± ( x, τ ) := lim h → h E τ n(cid:2) V ρ ± ( τ + h ) − V ρ ± ( τ ) (cid:3)(cid:2) X σ ( τ + h ) − X σ ( τ ) (cid:3)o + 12 h E τ n(cid:2) X ρ ( τ + h ) − X ρ ( τ ) (cid:3)(cid:2) V σ ± ( τ + h ) − V σ ± ( τ ) (cid:3)o ,a ρσ −± ( x, τ ) := lim h → h E τ n(cid:2) V ρ ± ( τ ) − V ρ ± ( τ − h ) (cid:3)(cid:2) X σ ( τ ) − X σ ( τ − h ) (cid:3)o + 12 h E τ n(cid:2) X ρ ( τ ) − X ρ ( τ − h ) (cid:3)(cid:2) V σ ± ( τ ) − V σ ± ( τ − h ) (cid:3)o . (7.27)Using the parallel transport equation (2.80), we then find a µ + ± = lim dτ → dτ E τ h d + ˆ v µ ± + Γ µνρ ˆ v ν ± d + x ρ + (cid:0) ∂ ν Γ µρσ + Γ µνκ Γ κρσ − µρκ Γ κνσ (cid:1) ˆ v ν ± dx ρ · dx σ + o ( dτ ) i = ∂ τ ˆ v µ ± + v ν + ∂ ν ˆ v µ ± + v ρσ + ∂ ρ ∂ σ ˆ v µ ± + Γ µνρ ˆ v ν ± v ρ + + (cid:0) ∂ ν Γ µρσ + Γ µνκ Γ κρσ − µρκ Γ κνσ (cid:1) ˆ v ν ± v ρσ + = ∂ τ ˆ v µ ± + ˆ v ν + ∇ ν ˆ v µ ± + ˆ v ρσ + ∇ ρ ∇ σ ˆ v µ ± − µνρ ˆ v ρσ + ∇ σ ˆ v ν ± − R µρσν ˆ v ρσ + ˆ v ν ± (7.28)40nd a µ −± = lim dτ → dτ E τ h d − ˆ v µ ± + Γ µνρ ˆ v ν ± d − x ρ − (cid:0) ∂ ν Γ µρσ + Γ µνκ Γ κρσ − µρκ Γ κνσ (cid:1) ˆ v ν ± dx ρ · dx σ + o ( dτ ) i = ∂ τ ˆ v µ ± + ˆ v ν − ∇ ν ˆ v µ ± + ˆ v ρσ − ∇ ρ ∇ σ ˆ v µ ± − µνρ ˆ v ρσ − ∇ σ ˆ v ν ± − R µρσν ˆ v ρσ − ˆ v ν ± , (7.29)where we allow for an explicit proper-time dependence of the velocity v ± ( X, τ ). For the secondorder parts we find a ρσ + ± = lim dτ → dτ E τ h d ˆ v ( ρ ± · dx σ ) + Γ ( ρ | κλ ˆ v κ ± dx λ · dx | σ ) + o ( dτ ) i = ˆ v ρκ + ∇ κ ˆ v σ ± + ˆ v κσ + ∇ κ ˆ v ρ ± (7.30)and a ρσ −± = lim dτ → dτ E τ h − d ˆ v ( ρ ± · dx σ ) − Γ ( ρ | κλ ˆ v κ ± dx λ · dx | σ ) + o ( dτ ) i = ˆ v ρκ − ∇ κ ˆ v σ ± + ˆ v κσ − ∇ κ ˆ v ρ ± . (7.31)Eq. (7.25) can now be rewritten as the stochastic Newton equation F µ ( X, τ ) = 12 m (cid:2) ˆ a µ + − ( X, τ ) + ˆ a µ − + ( X, τ ) (cid:3) , (7.32)where ˆ a µ = a µ + Γ µρσ a ρσ is the covariant form of a µ and F µ is a first order vector.There exists another representation of the stochastic Newton equation that is given by F µ ( X, τ ) = 12 m ( D + D − + D − D + ) X µ , (7.33)where the covariant diffusion operators D ± act on an arbitrary first order ( k, l )-tensor field A ( X, τ ) as, cf. Refs. [11, 21, 22], D ± A = (cid:20) ∂∂τ + ˆ v µ ± ∇ µ + ˆ v µν ± (cid:0) ∇ µ ∇ ν + R · µ · ν (cid:1)(cid:21) A, (7.34)where R · α · β A µ ...µ k ν ...ν l = k X i =1 R µ i αλβ A µ ...µ i − λµ i +1 ...µ k ν ...ν l − l X j =1 R λαν j β A µ ...µ k ν ...ν j − λµ j +1 ...ν l . (7.35)Using that v µν ± = ± ~ m g µν , eq. (7.34) can be rewritten as D ± A = (cid:18) ∂∂τ + ˆ v µ ± ∇ µ ± ~ m (cid:3) DG (cid:19) A, (7.36)where the Dohrn-Guerra Laplacian is defined by (cid:3) DG := g µν (cid:0) ∇ µ ∇ ν + R · µ · ν (cid:1) . (7.37)41 .3 Schr¨odinger equation The solutions of the stochastic differential equation (7.4) are stochastic processes. One canassociate a probability density to these stochastic processes, and derive a partial differentialequation for the evolution of this probability density. As argued in the introduction, the equationgoverning this evolution is the Schr¨odinger equation. Here, we present an explicit derivation.Using eqs. (7.17) and (7.18), we can rewrite eqs. (7.19) and (7.20) as ∂∂τ S ( x, τ ) = − m (cid:18) ∇ µ S ∇ µ S − ∇ µ R ∇ µ R − ~ (cid:3) R + 2 ~ A µ ∇ µ S + ~ A µ A µ + ~ R (cid:19) − U , (7.38) ∂∂τ R ( x, τ ) = − m (cid:18) ∇ µ S ∇ µ R + A µ ∇ µ R + ~ (cid:3) S + ~ ∇ µ A µ (cid:19) . (7.39)If we define the wave function Ψ( x, τ ) = e ~ ( R + iS ) , (7.40)we find that these equations are equivalent to the equation i ~ ∂∂τ Ψ = (cid:26) − ~ m (cid:20) ( ∇ µ + iA µ ) ( ∇ µ + iA µ ) − R (cid:21) + U (cid:27) Ψ . (7.41)This is a generalization of the Schr¨odinger equation to pseudo-Riemannian geometry. We notethat the Born rule is an immediate consequence: | ψ ( x, τ ) | = e ~ R ( x,τ ) = ρ ( x, τ ) (7.42)by the definition of R in eq. (7.8). In this section, we show that the generalization of the Schr¨odinger equation (7.41) imposes aconformal coupling of massive scalar particles to gravity. For this, we consider the Lagrangianof a free scalar field non-minimally coupled to gravity L ( φ, ∇ φ ) = − (cid:18) ∇ µ φ ∇ µ φ + m ~ φ + ξ R φ (cid:19) . (7.43)The field equation is given by the Klein-Gordon equation (cid:3) φ = m ~ φ + ξ R φ. (7.44)We can construct an explicitly proper time dependent field Φ, such thatΦ( x, τ ) = φ ( x ) e im ~ τ , (7.45)where x = ( t, ~x ) is a four-vector. Then Φ satisfies the generalized Schr¨odinger equation (7.41)with A µ = 0, U = 0 and conformal coupling ξ = . This result can be generalized in astraightforward manner to the cases A µ = 0 and/or U = 0.We conclude that stochastic quantization predicts that any scalar test particle must be con-formally coupled to gravity. It is expected that this result can be generalized to arbitrary scalarfields. However, proof of this latter statement can only be achieved within a field theory descrip-tion of stochastic quantization. 42 Discussion
In this paper, we have reviewed some aspects of second order geometry and stochastic quan-tization, and shown that the combination of the two leads to a consistent quantum theory onmanifolds. In addition, we have further developed second order geometry, and constructed thenotion of a Lie derivative in this framework. Furthermore, we have provided new results withinstochastic quantization. In particular, we have shown that a diffeomorphism invariant frame-work of stochastic quantization imposes a conformal coupling of massive spin-0 test particles. Itis expected that this result can be generalized to arbitrary scalar fields, but a proof of such ageneralization requires further study of a field theory framework.Since stochastic quantization can be formulated on (pseudo-)Riemannian manifolds, it isa natural approach to explore quantum gravity. However, in order to do so, a major hurdlemust still be overcome, which is a consistent extension to both bosonic and fermionic fieldtheories. Until now only a few specific bosonic examples have been studied in this framework,see for example Refs. [25–33], but no general formalism has yet been developed. The embeddingof stochastic quantization into second order geometry, as developed in this paper could helpguide the way towards such an extension. Particularly interesting in this respect are recentdevelopments in the study of Lagrangian dynamics on higher order jet bundles, see e.g. Refs. [57,58], as this is the natural extension of second order geometry to a field theory setting.There are several studies that can be performed within the stochastic quantization frame-work without going to a field theory description or to dynamical backgrounds. The stochasticdifferential equation (7.4) allows to solve and simulate the motion of quantum mechanical spin-0test particles charged under scalar and vector potentials in any geometry. Such a study wouldbe particularly interesting when performed in black hole geometries. One can then calculate theprobability that a particle hits the singularity or escapes the black hole. Furthermore, one cancalculate the expected proper time until one of these events occurs. Also, higher moments suchas the variance for these events can be calculated. Such calculations could provide microscopicinsights into Hawking radiation and black hole thermodynamics.In this paper, we have restricted ourselves to time-like processes with positive mass. Aformulation for space-like processes can be obtained by considering imaginary masses and byreplacing the proper time with the proper distance. However, a theory for massless particles onnull-like surfaces is not easily obtained from the theory presented in this paper, and deservesfurther study.There are many other issues that deserve further exploration within the stochastic framework.For example, as discussed in the introduction, there is no consensus yet on the resolution ofWallstrom’s criticism. Moreover, the notion of spin in stochastic quantization is only partiallyunderstood, see e.g. Refs. [11, 20, 34]. In this paper, we have focused on scalar particles, inthe presence of commuting spin-0 and spin-1 fields and gravity. Extensions to fermions, non-commuting potentials and higher spin fields would be interesting to investigate.Furthermore, the formulation of stochastic quantization presented here was entirely in aposition representation. Further investigation of the dual picture in terms of momenta deservesfurther exploration. Early considerations along these lines can for example be found in Ref. [59].Another open question is whether stochastic quantization can be formulated on complexmanifolds instead of real manifolds. An argument for such a construction is that the wavefunction resembles the probability density of a complex random variable Z = X + iY with dZ = ( V + iU ) dτ . Discussions along these lines can also be found in Ref. [60]. Related to this In stochastic quantization, geodesic incompleteness of the space-time does not imply that the particle endsup at the singularity. One should study the Brownian completeness of the geometry instead, see e.g. Section 5 inRef. [56].
43s the question whether the function R can be interpreted as an action for the background fieldin a Wick rotated version of the theory. The action S would then be related to the probabilitydensity for the coordinates Y .Finally, the presence of an osmotic velocity in stochastic quantization could provide newinsights in the nature of dark matter. In this respect, it is worth noticing that the kinetic energyin stochastic quantization does not only contain the classical kinetic energy given by m g µν v µ v ν ,but also the osmotic energy of the background field given by m g µν ˆ u µ ˆ u ν . It is expected that thenotion of osmotic energy is also present in a field theoretical extension of stochastic quantization.In such an extension it will take the shape of the kinetic term of additional fields that only interactgravitationally with other fields. This suggests that the osmotic energy could be interpreted asdark matter.We conclude that stochastic quantization is an interesting framework, that deserves furtherexploration. We are currently investigating several aspects of the theory along the lines mentionedabove, and hope to report on it elsewhere. Acknowledgments
This work is partially supported by a doctoral studentship of the Science and Technology Fa-cilities Council. I would like to thank Joshua Erlich for interesting discussions on stochasticquantization. Furthermore, I would like to thank Xavier Calmet for helpful comments on themanuscript.
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