Bessel Process and Conformal Quantum Mechanics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Bessel Process and Conformal Quantum Mechanics
M. A. Rajabpour a ∗ a Dip. di Fisica Teorica and INFN, Universit`a di Torino, Via P. Giuria 1, 10125 Torino,ItalyNovember 16, 2018
Abstract
Different aspects of the connection between the Bessel process and the conformalquantum mechanics (CQM) are discussed. The meaning of the possible generalizationsof both models is investigated with respect to the other model, including self adjointextension of the CQM. Some other generalizations such as the Bessel process in the widesense and radial Ornstein- Uhlenbeck process are discussed with respect to the underlyingconformal group structure.
Keywords : Bessel Process, Conformal Quantum Mechanics, Self adjoint ExtensionPACS number(s):
The Bessel process is one of the building blocks of stochastic processes because of its appli-cations and also its simplicity and richness. The Bessel process describes the norm of theBrownian motion in arbitrary dimension [1, 2]. This process describes the movement of anarbitrary point on the real line in the Schramm-Loewner evolution [3]. It is also importantin the probabilistic description of some financial markets specially Cox, Ingersoll and Ross(CIR) model [4]. Since the Bessel process is connected to the movement of a free random walkin arbitrary dimension it could be connected to the quantum mechanics of the free particle.The interpretation of quantum mechanics as the classical stochastic equation is a long story ∗ e-mail: [email protected] To define Bessel process we start with the primary motivation for definition of the process.It is just the radial part of δ -dimensional Brownian motion, i.e, R t = q B ( t ) + ... + B δ ( t ).Using Ito’s formula one can write the following stochastic equation for the Bessel process dR t = δ − R t dt + dB t (2.1)where B t is the one dimensional Brownian motion. It is easy to see that the above process hasscaling property, i.e. if R t is a Bessel process with starting point x then the process c − R c t is also Bessel process with starting point at x/c for positive c . In this level one can considerthe above equation with the arbitrary real δ . The Bessel process with positive δ was studiedextensively in the literature [1, 2]. Before establishing the properties of Bessel process it isworth mentioning some properties of Brownian motion in arbitrary dimension.It is well known that Brownian motion in δ ≤ δ >
2. Withtransient we mean that if δ > t →∞ | B t | = + ∞ and the process isrecurrent if the set { t : B t ∈ U } be unbounded for all the sets D in R δ ; in other words theprocess X t said to be recurrent if P x ( T y < ∞ ) = 1 for all x, y ∈ D where T y is the hitting time,the first time at which the process hits point y , and P x ( T y < ∞ ) is the probability of havingfinite hitting time of the point y for the process with starting point at x . Another interestingproperty of Brownian motion is related to the probability of meeting two Brownian paths witharbitrary starting points in finite time. Two Brownian paths will meet each other if δ < δ ≥ δ ≥
2, regular if 0 < δ < δ ≤
0. One cansummarize also the following properties I : for δ > ,II : for δ ≤ ,III : for δ ≥ , V : for 0 < δ < , origin could be a killing point or reflecting point . The same properties are true for the squared Bessel process defined by Z t = R t whichsatisfies the following equation dZ t = δ ( dt ) + 2 q | Z t | dB t . (2.2)Using Ito’s formula the generator of squared Bessel process is L f ( x ) := 2 xf ′′ ( x ) + δf ′ ( x ) . (2.3)Then one may introduce the Green’s function G λ ( x, y ) as the Laplace transform, with respectto time, of the transition density, the density of finding the process starting from x at y , ifof the process G λ ( x, y ) = Z ∞ e − λt p ( t, x, y ) . (2.4)The Green’s function satisfies the following equation L G λ ( x, y ) − λG λ ( x, y ) = 0 . (2.5)The solution may be factorized as follows G λ ( x, y ) = w − λ ψ λ ( x ) φ λ ( y ) if x ≤ yw − λ ψ λ ( y ) φ λ ( x ) if x ≥ y , (2.6)where w λ is the Wronskian w λ := ψ ′ λ ( x ) φ λ ( x ) − ψ λ ( x ) φ ′ λ ( x ). Since in the case of 0 < δ < boundarycondition or reflecting boundary condition. Using the above boundary conditions one couldfind the following solutions ψ λ ( x ) = x − ν I ν ( √ λx ) if δ ≥ < δ < x − ν I − ν ( √ λx ) if δ ≤ < δ < φ λ ( x ) = x − ν K ν ( √ λx ) , (2.8)where I ν and K ν denote the modified Bessel functions with index ν = δ − . It is easy to seethat w λ = . Using the above solutions one can write the transition density for all δ > In some literatures the word absorbing was used p ( t, x, y ) = 12 t ( xy ) − ν e − x + y t I ν ( √ xyt ) if x > , (2.9) p ( t, x, y ) = 2(2 t ) ν Γ(1 + ν ) e − y t if x = 0 . (2.10)The transition density for 0 < δ < p ( t, x, y ) = 1 t ( xy ) − ν e − x + y t I − ν ( √ xyt ) . (2.11)Similar results could be calculated for the the Bessel process, in this case the generator hasthe following form L f ( x ) := 12 f ′′ ( x ) + 12 x ( δ − f ′ ( x ) . (2.12)Since this process is just the square root of squared Bessel process one can get the transitiondensities in this case by just the transformations x → x and y → y . For example for thereflecting origin we will have p ( t, x, y ) = 12 t y ( xy ) − ν e − x y t I ν ( xyt ) . (2.13)One could get the same answer by the method which is more familiar for physicists and thatis by using the Fokker-Planck equation which has the following form for the Bessel process ∂ t p ( t, x, y ) = 12 ( ∂ x − ∂ x νx ) p ( t, x, y ) . (2.14)One can do the transformation p ( t, x, y ) = x ν + Q ( x, y, t ) following by the Laplace transformand get the equation − ∂ x Q λ + ν − / x Q λ = − λQ λ . (2.15)where Q λ is the Laplace transform of Q ( x, y, t ). The above equation is just the modifiedBessel equation with the modified Bessel functions as solutions. It is easy to see that theabove eigenvalue problem by the change of variable S λ = x − ( ν +1 / Q λ is equivalent to ( − ∂ x − ( νx ) ∂ x ) S ( x ) = − λS ( x ) which the operator is the same as the generator of Bessel processstated before.So far we just addressed the Bessel process with natural boundary conditions in originand infinity but it is also possible to investigate squared Bessel process with δ > x ≤ δ ≤ δ = 0 the processwill reach zero and stays there. For a squared Bessel process with δ ≥ x ≤ − Z − δ − x process, with starting point at − x , until ithits origin and after that it behaves like Z δ . Similar relations are valid for the process with δ ≤ x > − Z − δ − x .It is worth to mention that since for negative dimensions the squared Bessel process becomenegative the square root of it, which is Bessel process, will become purely imaginary and soone could be careful about extending the results to the Bessel process, however up to thetime that the process is positive one could define the Bessel process as well as a real process.To complete the discussion we give the transition density of the squared Bessel process withnegative dimension given in [12] for starting value x > p ( t, x, y ) = k ( x, y, δ, t ) e − x + | y | t Z ∞ (1 + w ) − δ w − δ/ exp ( − t ( xw + | y | w )); k ( x, y, δ, t ) = − δ δ Γ [ − δ ] x − δ | y | − (1+ δ ) t δ − . (2.16)Another interesting process related to the Bessel process is the time reversed Bessel pro-cess. If R t be a Bessel process with starting point on the positive real line with dimensionsmaller than two then the time reversed Bessel process, defined by ( R ( T x → ) − s , s ≤ T x → ),has the same law with a Bessel process ˆ R s , s ≤ ˆ L → x starting from origin with ˆ δ ≡ − δ andˆ L → x ≡ { sup t | ˆ R t = x } . In the above notation T x → y ≡ inf { t | R t = y } is the first time thatthe process starting at x hits the point y . To make it more clear consider the ensemble of allBessel paths that ended up when they hit the origin in the first time, then consider all thesepaths in the reverse time, the laws of these two processes are the same. The other way tosay the above statement is as follows: For every bounded function F one can write E δ [ F ( R ( T x → ) − s , s ≤ T x → )] = E − δ [ F ( ˆ R s , s ≤ ˆ L → x )] , (2.17)where F is a function on the set of realizations of the process. The above equality relatesBessel process with dimension δ to Bessel process with dimension δ ′ = 4 − δ . One can writethis duality by defining κ = − δ as κ + κ ′ = . Using the equality one could easily see that n = sin( πκ ) = sin( πκ ′ ). The above equality is valid for all real values of δ . It is also possibleto write the equality as ν + ˆ ν = 0, then one could say that these two processes have the same Q ( x, y, t ) if we work in the positive range of dimensions of Bessel process. In fact we willsee in section two that these two different Bessel processes are related to the same conformal6uantum mechanics. It is also easy to see that δ = 2 is self dual and δ = 4 is critical whichbeyond that our reverse time process has negative dimension. The above equality is alsouseful to get the law of certain first hitting times of Bessel process [12]. In order to introduce conformal quantum mechanics we follow the same strategy that wehad in the previous section, we start from the free particle in δ dimensions. The Schr¨odingerequation for a free particle in radial coordinates has the following form − r δ − ∂∂r ( r δ − ∂ϕ ( r ) ∂r ) = Eϕ ( r ) , (3.1)or after differentiation one could write the following Bessel generator eigenvalue equation − ∂ ϕ ( r ) ∂r − δ − r ∂ϕ ( r ) ∂r = 2 Eϕ ( r ) . (3.2)As it is evident a similar equation was derived for the Bessel process in Laplace space in theprevious section. By ϕ ( r ) = r − ( ν +1 / Q ( r ) and extending the range of δ to all real values wewill have the following Hamiltonian for conformal quantum mechanics H = 12 ( p + ν − / r ) , (3.3)where p = i ∂∂r and [ r, p ] = i at the quantum level. The above quantum mechanics as a singularquantum mechanics was discussed extensively soon after the discovery of quantum mechanics[13, 14] and references therein. It is easy also to get the same quantum mechanics for twofree particles moving in δ dimensions with r interaction by just forgetting the momentumof the center of mass term and going to the radial part of the spherical coordinates. Theconformal quantum mechanics has many interesting properties. Before investigating differentsolutions of the above system we study some symmetries of the system. The Hamiltonianat the level of one dimensional quantum mechanics with usual inner product for Q ( r ) issymmetric with respect to ν → − ν but at the level of ϕ ( r ) is not symmetric with respectto the same transformation. The action corresponding to the Hamiltonian (3.3) is invariantunder the following transformations t ′ = at + bct + d ,r ′ ( t ′ ) = r ( t ) ct + d with ad − bc = 1 . (3.4)7he above transformations are the conformal transformations in 0 + 1 dimensions [7]. Thebasic transformations are time translation, dilation and special conformal transformationwith the following Noether charges t ′ = t + b, H = H = 12 ( p + ν − / r ); (3.5) t ′ = α t, D = tH −
14 ( rp + pr ); (3.6) t ′ = tct + 1 K = t H − t ( rp + pr ) + 12 r . (3.7)The above generators verify the algebra of the conformal group SO (1 ,
2) which is[
H, D ] = iH, [ D, K ] = iK, [ H, K ] = 2 iD. (3.8)Using the new definitions L = ( H + K ) and L ± = ( H − K ± iD ) one can write thealgebra in the more familiar form[ L , L ± ] = ∓ L ± , [ L +1 , L − ] = 2 L . (3.9)Explicit dependence of K and D on t will guarantee their conservations. The importantthing to mention is that the argument is not considering the most general case because wealready knew that the action is invariant up to a total derivative which could be non zeroin the presence of the boundary in the origin. We will discuss both conformal invariant andanomalous case. This will be more clear when we discuss the self adjoint extension of thequantum mechanics. Another important thing to mention is although it seems that we foundthree conservation laws for our two-dimensional phase system, it is not difficult to check thatthey are in fact related by the following relation HK + KH − D = ν − − , (3.10)which is also the Casimir operator of the group. In order to present different aspects of theabove quantum mechanics we introduce another variable g = ν − . (3.11)Our quantum mechanics has different properties with respect to the value of g , some of whichwe will summarize in the following. For an arbitrary value of g it is easy to show that if onecould find one bound state with energy E then scaling the position by an arbitrary factor α it is easy to construct a new solution with energy α E , which means that if there exist8ny bound states then there is a bound state for every negative energy. This is a directconsequence of the scaling property of this model. The same story is true for the positiveenergy solutions of the model which means that we have just planar waves with all the possiblepositive energies which was also used to obtain equation (2.15). In the previous section weconsidered lim r → ϕ ( r ) = 0 for the killing boundary condition and for the reflecting casewe had lim r → ∂ϕ ( r ) ∂r = 0. In the level of conformally invariant quantum mechanics we willignore the negative energy solutions which means that our Hilbert space is made by wavefunctions corresponding to continuous positive energy solutions. The corresponding boundarycondition also will be discussed in the end of this section.The Hamiltonian could have other solutions dependent on the value of the coupling g .Firstly we should mention that the Laplacian operator that was the starting point is selfadjoint if we consider it in the whole space. But one could extract other solutions by removingthe origin from the domain and considering the self adjoint extension of the operator in thenew domain. Of course existence of the extension is dependent on the value of g or in otherwords to the corresponding dimension. All the necessary aspects of theory of self adjointextension were discussed in the appendix A. In the case of our quantum mechanics thepossible extensions were discussed in [15, 16, 17].For g ≥ the Hamiltonian is self adjoint with removed origin now and has a scatteringsector with the following solutions ϕ ( r ) = ( √ Er ) J ν ( √ Er ) or ( √ Er ) Y ν ( √ Er ) , (3.12)where J and Y are the Bessel functions. For ν ≥ J can be considered because inthis case Y at the origin is infinity and for ν ≤ − Y has desired property. For ν ≥ δ ≥ r → ϕ ( r ) = 0 and can not be extended.For g < since the Hamiltonian is not self adjoint one can find a required self adjointextension. It is better to distinguish between the domain − ≤ g < and g < − . First wediscuss the domain − ≤ g < which is equivalent to − < ν < < δ <
4. In this rangethe Hamiltonian requires a self adjoint extension with the self adjoint parameter z which isresponsible to map two deficiency subspaces by the unitary map e iz . The important point tomention is that although the Hamiltonian in the domain D ( H ) ≡ { ϕ (0) = ϕ ′ (0) = 0 } is notself adjoint, it is still Hermitian. 9f we consider H † as the adjoint of H , with the same differential representation as H , thenfrom the Von Neumann’s theory of deficiency indices we know that the deficiency subspaces K ± are made by the square integrable solutions of the equation H ∗ φ ± = ± iφ ± in the desireddomain. In our case the solutions are φ + ( r ) = r H ν ( re i π ) , (3.13) φ − ( r ) = r H ν ( re − i π ) , (3.14)where H ν is the Henkel function and both of φ ± ( r ) are square integrable in the half line.Using the above solutions one could argue that the Hamiltonian is self adjoint in the domain D z ( H ) = D ( H ) ⊕ { u ( φ + ( r ) + e iz φ − ( r )) } where u is an arbitrary complex number. Tofind the valid boundary condition we need ψ to be in the self adjoint domain. ConsiderΦ = φ + ( r ) + e iz φ − ( r ) then ψ is in the self adjoint domain if < Φ | Hψ > = < H Φ | ψ > orlim r → [Φ ∗ dψdr − ψ d Φ ∗ dr ] = 0 . (3.15)To check this we need the behavior of Φ close to the originΦ( r ) → i sin( πν ) [ Ar ν + + Br − ν + ] , (3.16)where A = e − i πν − e i ( z + πν ) ν Γ(1 + ν ) ,B = e i ( z + πν ) − e − i πν − ν Γ(1 − ν ) , (3.17)for ν = 0. Using the above relations one can write the equation (3.16) for the boundarycondition as follows( Ar ν +1 / + Br − ν +1 / ) dψdr − ( A ( ν + 12 ) r ν − / + B ( 12 − ν ) r − ν − / ) ψ → . (3.18)The equation for ν = is like B dψdr | = Aψ (0). Actually ν = is very interesting becausefirstly it corresponds to δ = 3 and also because it describes the possible boundary conditionsfor the free particle in the half line. The simplicity of the results helps us to investigate thepossible meanings of the above self adjoint extension with respect to the Brownian motionand the Bessel process which we will discuss in more detail in the next section. The nextimportant case which deserves separate calculations is ν = 0. In this case the function Φ hasdifferent properties near the origin. Using φ + ( r ) = φ ∗− ( r ) andΦ( r ) → iπ r ln( r ) + [ 12 + 2 iπ ( γ − ln 2)] r + e iz cc = ( A + A ∗ e iz ) r ln( r ) + ( B + B ∗ e iz ) r . (3.19)10here γ is the Euler constant and cc is the conjugate of the first term. One can find thefollowing boundary condition A r ln( r ) dψdr − √ r [ A (1 + ln r B ψ → , (3.20)where A = A + A ∗ e iz B = B + B ∗ e iz . (3.21)One can extend the results also for the case with g < − which is related to pure complex ν .In this case we have still a one parameter self adjoint extension and all of the results are stillvalid if we put imaginary ν in to the formulas, in particular in to the (3.18). The importantpoint to mention is in fact that by introducing the self adjoint parameter we are classifyingall the possible physical boundary conditions of our system. In our case we should mentionthat after the self adjoint extension our Hamiltonian is not necessarily scale invariant andso could have a negative ground state which is in fact the case here. Of course the groundstate depends on the self adjoint extension parameter. The energy and wave function forour Hamiltonian were discussed before in [16] and references therein, and the results are asfollows.The Hamiltonian does not have any bound state for g ≥ but admits one bound statein the range − ≤ g < . The wave function of the bound state up to the normalizationconstant has the following form ψ ( r ) ∼ r K ν ( p − E ν r ) , (3.22)with energies E ν =0 = − ( sin( z + πν )sin( z + πν ) ) ν , (3.23) E ν =0 = − exp π z . (3.24)For the case g < − we have infinite discrete bound states [13] with the same wavefunctionas above but with imaginary ν . To get the energies we just need to put the wave function inthe corresponding boundary condition which yields E = − AB ) ν exp { ν ) − nπ } , for n ∈ Z . (3.25)It is worth to emphasize again that since after the self adjoint extension the action is not scaleinvariant we do not need to worry about having a bound state because it does not enforceany other bound state by the scaling argument.11ne could summarize this section as follows: we introduced conformal quantum mechanicsas the quantum mechanics of the free particle in δ dimensions. Constraining the domain ofthe quantum mechanics to the space without origin forces new boundary conditions to thesystem. By these boundary conditions for some values of g we need to find a self adjointextension of the Hamiltonian. The method of extension helps to find all the relevant boundaryconditions and so the physics of the model is related drastically to the self adjoint extensionparameter. The extended Hamiltonian which is not scale invariant admits discrete boundstates. Since after the self adjoint extension our conformal symmetry has some anomalies,may be the expression conformal quantum mechanics is a bit confusing but because of our firstmotivation we will keep using it. Different aspects of this anomaly were discussed extensivelyin [18, 10, 19, 20] from the re-normalization group point of view. The above results couldbe also explained from the framework of two free particles moving in δ dimensions. Theself adjoint extension is just giving boundary conditions corresponding to the time that twoparticles meet each other. For example, bound states of the above system could be seen asthe bound states of two particles which behave like one particle. In the next section we givesome hints on how one can translate the above results in the Brownian motion language. In this section we are interested in discussing different features of the similarity betweenBrownian motion in δ dimension and free particles from a quantum mechanics point of viewand also similarities between Bessel process and CQM.The first remark is for the Bessel process with positive dimension. We had origin andinfinity as the natural boundary conditions which is also the case for our CQM. In additionbefore extending the Hamiltonian we have a scale invariant system which is the case also inthe Bessel process. Just as we remarked in the previous section, keeping the scaling symmetryof the CQM gives us an unbounded continuous spectrum which is also true if we try to findthe transition density of the Bessel process. To make more clear the connection between twomodels one can use the following relation between the transition density of the Bessel processand the path integral of CQM p ( x, y, t ) = Z q ( t )= yq ( t )= x [ dq ( τ )] exp[ − S ( q ( τ ))] (4.1)12 ( q ) = Z
12 ( ˙ q + 1 − δ q ) dt (4.2)It is not difficult to see that by a transformation p ( t, x, y ) = x ν + Q ( x, y, t ) one can go fromthe above action to the familiar conformal action that we discussed in the previous section.By the above discussion it seems that the self adjoint extension is something beyond ournormal understanding of the Bessel process. It is not difficult to see that it is related tothe different possible boundary conditions for the Bessel process at the origin. Different selfadjoint extensions are related to different boundary conditions or to different measuring of thepaths of the Bessel process. To make it more rigorous, by Bessel process we mean some kindof generalized Bessel process which is related to the diffusion of Brownian motion in the spacewithout the origin. From the two Brownian motions point of view naively one could arguethat since we have a self adjoint extension just for δ < δ < R δ is by consideringthe delta function potential at the origin. Of course this will work just for the integer δ .The cases δ = 1 , , δ ≥ V ( r ) = vδ ( r ) we13an use V ( r ) = c d v πR d − δ ( r − R ) , with c = 1 , c = 12 ; (4.3)for dimensions d = 2 ,
3. The corresponding limit is R →
0. In the case of the square wellwe have V ( r ) = − D Θ( R − r ), where Θ is the Heaviside function, D is a coonstant andthe limit is the same. Since these regularization procedures were discussed extensively, weignore the details but we explain the strategy, for more details see [21]. Using the aboveregularized delta potentials firstly we should solve the problem for the regularized potentialby considering the continuous wave function at r = R . This matching leads to equationsbetween different parameters including v , R and energy. Then by defining the re-normalizedcoupling ˜ v as a function of R and v , one could derive the energy and wave function of thesystem exactly as we found in the previous section. For example for the three dimensionalcase we have by definition v = v + πR and after matching the wave functions we will have thefollowing equations for the bound state energy E b and the phase shift of the s -wave scatteringsector p E b = 2 π ˜ v , (4.4)tan( δ ) = − ˜ vk π . (4.5)Similar results for the two dimensional delta function were given in [21].This regularization procedure will not work for dimensions d ≥ η , ψ (0) = − η π dψdr | r =0 . (4.6)14or positive energy E = k / ψ ( r ) = 1 r (sin( kr ) + tan( δ ) cos( kr )) , (4.7)where δ is the phase shift corresponding to the s -wave. It is not difficult to see that for theattractive delta function using equation (4.6) and (4.7) one could get η = ˜ v . This matchingcan not be done for the repulsive case because we already know that the repulsive deltafunction is trivial, but the self adjoint extension with η < δ as the stochastic process in the presence of theregularized delta function. However ,writing a stochastic equation for the radial Brownianmotion with one removed point is not obvious and needs more investigation. Of course thedefinition of a stochastic process for the generic case with arbitrary δ or ν is more difficult.Although the boundary condition for δ = 3 or ν = , which is equivalent to the half line freequantum particle, is very simple, it is enlightening. One could see the boundary condition(4.6) from the stochastic process point of view at one dimension as follows: η = 0 is theDirichlet boundary condition which is an absorbing boundary condition. The particle willbe absorbed by the origin after hitting that. This also corresponds to the conventional freeHamiltonian with scale invariance. From the two particle point of view it is like absorptionof one particle by the other one when they touch each other. The Dirichlet boundary caseis just reminiscent of the equality of the 3 dimensional Bessel process with the Brownianmotion on the half line with absorbing boundary condition.The other extreme case is the Neumann boundary condition η → −∞ which is a reflectingboundary condition at the origin for the Brownian motion. Other negative values of η corre-spond to a mixing of Dirichlet and Neumann boundary conditions. This correspondence wasdiscussed in detail in [24, 25, 26] and the Green function has the following form for arbitraryvalue of the self adjoint extension G η ( x, y, t ) = G F ( x − y, t ) + G F ( x + y, t ) + 4 πη Z ∞ dwe πη w G F ( x + y + w ) η ≤
0; (4.8) G η ( x, y, t ) = G F ( x − y, t ) + G F ( x + y, t ) − πη Z ∞ dwe − πη w G F ( x + y − w )+ 4 πη e i π tη e − πη ( x + y ) η ≥
0; (4.9) G F ( x − y, t ) = 1 √ πit e i ( x − y ) / t . (4.10)15or the special cases, Dirichlet and Neumann the results are as follows G η =0 ( x, y, t ) = G F ( x − y, t ) + G F ( x + y, t ); (4.11) G η →−∞ ( x, y, t ) = G F ( x − y, t ) − G F ( x + y, t ) . (4.12)One can use the above equations to get the Green’s function of the Brownian motion in threedimensions when the origin is removed. Of course one could get the Green’s function for thegeneral case by using an orthogonal eigenvalue expansion for the self adjoint operator. TheGreen’s function with respect to the solutions of the Hamiltonian has the following form G η ( x, y, t ) = Z ∞ dke − iE k t ϕ ( y ) ϕ ∗ ( x ) . (4.13)In the case of g < since we have also a bound state we need to add new terms coming fromthe discrete contribution of this states by adding P b ( e iE b t ψ ( r ) ψ ∗ ( r )) to the Green’s function,where E b is the bound state energy. The important conclusion is that one could also considerthe above Green’s function as sort of a generalized Bessel process in δ dimensions. In factone can get the transition density for the above processes by just Wick rotation. The abovegeneralization of Bessel process has not been appeared in the mathematical literature. Fromthe stochastic process point of view one can derive the above solutions by considering thelocal time of the process [1]. The definition of the local time of the path ω at the point a isas follows t l ( a ) := 12 lim ǫ → Z T x + ǫ ( B s ) ds, (4.14)where x + ǫ ( B s ) is the indicator for the time that the process is in the interval [0 , x + ǫ ].One could naively write the above equation as an integral over a delta function as t l ( ω,
0) = R T δ ( x ( t )) dt . Using the above equation the extended transition density for δ = 3 is just theexpectation value of exp ( − πη t l ). The corresponding stochastic process for the free particle inthe half line is called elastic Brownian motion [1]. To the best of our knowledge the problemhas not been discussed by the mathematicians for the generic case. For the Bessel processthat we discussed in the second section one just needs to consider B = 0 in the equation(3.18). In this case we will recover conformal symmetry for our process again. It will bealso interesting if one could get the results for the Bessel process with negative dimension byusing the quantum mechanics of a free particle, in particular the equation (2.16).16 Generalization of Bessel Process and CQM
In this section we want to present some possible generalizations of the Bessel process withthe well known quantum mechanical counterparts such as, Bessel process with constant drift,Bessel process in the wide sense, Cox, Ingersoll and Ross (CIR) model and Morse processwhich are related to a Coulomb potential, conformal quantum mechanics, radial harmonicoscillator and Morse potential respectively. Of course not all of the above processes admitconformal symmetry; they are reasonable perturbations of the conformal invariant case.
The definition of the Bessel process with constant drift is given by the following stochasticprocess dR µt = ( δ − R t + µ ) dt + dB t . (5.1)From the Feller classification the boundary at the origin has the same classification as for thestandard Bessel process, the infinity is a natural boundary condition which is attracting if µ is positive and non-attracting for a negative drift. The generator of the process is L f ( x ) := 12 f ′′ ( x ) + ( 12 x ( δ −
1) + µ ) f ′ ( x ) . (5.2)One could map the solution of the equation (2.5) to the Schr¨odinger equation of the Coulombpotential. To do so we need the Liouville transformation R λ ( r ) = 2 ( r √ δ − e µr √ G λ ( r √ , (5.3)where r = √ x . Using the above transformation one could write the equation (5.2) as d R λ dr + ( cr − l ( l + 1) r ) R λ = − ER λ , (5.4)where E = − λ + µ , l = δ − and c = − µ δ − √ . For integer δ the equation is related to theenergy levels of a Hydrogen atom. The different aspects of the equation (5.4) were discussedin [27] including the transition density for the different values of µ and δ , here we just mentionsome properties. The Bessel process with constant drift admits a discrete spectrum for allnegative values of µ which is expectable for those people who are familiar with the Hydrogenatom. For 0 < δ < δ > µ for dimensions in the range 0 < δ <
1. In this case δ = 1 is the critical dimensionfor admitting discrete energies or not. Another very important generalization of the Bessel process is by considering a Brownianmotion in δ dimensions with a drift of magnitude µ ≥ , ∞ ). It was shown in [29] that the generator of this processis L f ( r ) := 12 f ′′ ( r ) + ( 12 r ( δ −
1) + h − ν ( µr ) dh ν ( µr ) dr ) f ′ ( r ) , (5.5)where h ν ( x ) = ( 2 x ) ν Γ(1 + ν ) I ν ( x ) , ν = δ − . (5.6)The transition density of this process is related to the transition density of the Bessel process p δ ( t, x, y ) as follows p δ,µ ( t, x, y ) = e − µ t/ h − ν ( µx ) p δ ( t, x, y ) h ν ( µy ) . (5.7)The above process has an interesting time inversion property. It was shown in [28], see also[30] that if X t with t ≥ tX t with t > α by demandinga Markov process with homogenous t α X t . They can be obtained by just considering Besselprocesses in the wide sense with appropriate power. The Hamiltonian corresponding to thisprocess after transformation p δ,µ ( t, x, y ) = r ν +1 / h ν ( µr ) Q ν ( µr ) is as follows H = 12 ( p + V ( r )) V ( r ) = ν − / r + µ , (5.8)where we used the equality µ = µ + µ r ( δI ν ( µr )+ µrI ν ( µr )) I ν ( µr ) . Since the third term does nothave any singularity it means that the general aspects of the above quantum mechanics arethe same as the Bessel process, in particular it will have a self adjoint extension for the range18 <
1. This equality means that basically the above two distributions have similar spectrumand also it is a very simple proof for the equation (5.7). One could use all the previoussolutions for the self adjoint Hamiltonian to get the solutions in this case. The consistency ofthe inner product is coming from the Doob’s h -transform. For the case of Brownian motionwith negative drift − µ one just needs to replace the modified Bessel function I ν ( µr ) with K ν ( µr ). The same as the Bessel process case here also one can generalize the Bessel processin the wide sense by considering all the possible self adjoint solutions of the Hamiltonian [ ? ]and then using the Doob’s h -transform. It will be really interesting to study this generalizedcases with respect to the time inversion. The Cox, Ingersoll and Ross (CIR) model [4] or radial Ornstein- Uhlenbeck process is widelyused for interest rate framework such as some stochastic volatility models [31]. The definitionof the CIR family of diffusions is by the following equation dN t = ( a − bN t ) dt + c q | N t | dB t , (5.9)with N ≥ a ≥ c > b is an arbitrary real number. It is not difficult to see that a = δ , b = 0 and c = 2 is the squared Bessel process. A CIR process can be represented interms of the squared Bessel process as follows N t = e − bt Z ( c b ( e bt − , (5.10)where Z t denotes the squared Bessel process of dimension δ = ac . The above equation canbe checked by using Ito’s formula for deterministic time transformation of a squared Besselprocess. Using the above connection one can easily classify the properties of the origin as aboundary for the process; we just need to consider ac as the dimension of the correspondingBessel process. To make the connection with the radial harmonic oscillator let’s first mapthe CIR process to the square root of it M t = p | N t | by using the Ito’s formula as follows dM t = (( a − c M − b M ) dt + c dB t . (5.11)One can easily map the above process to the radial harmonic oscillator by the followingHamiltonian H = 12 [ p + ω x + kx + e ] . (5.12)19here ω = b , k = ( a − c )( a − c −
1) and e = − b ( a − c ) − b . The above Hamiltonian isjust the radial part of the harmonic oscillator in δ dimension. The above calculation showsthat by deterministic time change one can go from scale invariant Bessel process to non-scale invariant process with stationary solution. This is a hint to believe that it should bepossible to move from a conformal quantum mechanics to a radial harmonic oscillator bytime translation. This is in fact natural and was done long time ago in [7]. The strategy isas follows: Firstly one could write SO (2 ,
1) generators in a more familiar form by definingnew generators as follows S = 12 ( 1 a K − aH ) ,R = 12 ( 1 a K + aH ) , (5.13)where a is a constant and the commutators are as follows[ D, R ] = iS, [ S, R ] = − iD, [ S, D ] = − iD. (5.14)The important point is that the operators D and S correspond to hyperbolic non-compacttransformations and R is the generator corresponding to a compact rotation. Since all of theabove operators are the invariants of the action one could define a generic operator as G = uH + vD + wK, (5.15)as a constant of the motion. Of course it will correspond to compact rotation in threedimensions if we consider ∆ = v − uw <
0. This will be important to get a theory withreasonable time evolution. From now on we will just consider this case. The simplest exampleof this kind of operators is R with ∆ = −
1. The action of the operator G on a wave functionis as follows G | Ψ( t ) > = i ( u + vt + wt ) ddt | Ψ( t ) > (5.16)which by time transformation could be written as G | Ψ( τ ) > = i ddτ | Ψ( τ ) > (5.17) τ = 4 w √− ∆ { arctan( 2 wt + v − ∆ ) − arctan( v − ∆ ) } . (5.18)In the new parametrization one could think about G as the new time translation and thusthe new Hamiltonian. After some algebra one could write the following Hamiltonian˜ H = 12 [ p − ∆4 x + gx ] , (5.19)20s the most general possible Hamiltonian that could be extracted from the SO (2 ,
1) groupwith applicable time translation. To compare with the CIR model one could write ∆ = − b and g = k which is an indication to believe that the corresponding time translation in a Besselprocess is related to a compact rotation in a three dimensional space with metric ( − , − , xy , yz and zx are R , D and S respectively. Differentaspects of the self adjoint extension of the above Hamiltonian were discussed in [32] andreferences therein and they are quite similar to the case without harmonic potential. In this subsection for the sake of completeness we want to discuss briefly another relatedphysical model, Schr¨odinger equation with Morse potential [33]. This potential is exactlysolvable and has many applications in molecular physics [33, 34]. This system is related tothe quantum mechanics of the radial Harmonic oscillator. One can find the Morse potentialby just making the variable change u = − x in the Schr¨odinger equation, Hψ ( x ) = Eψ ( x )with Hamiltonian (5.12). Then we will have the following Schr¨odinger equation with thecorresponding Hamiltonian H m φ ( u ) = E m φ ( u ) where H m = 12 p + ω e − u − E − e/ e − u , E m = −
14 ( 18 + k . (5.20)The above equality means that the Morse potential is just the canonical transformation,i.e. u = − x and p u = − xp x , of the radial Harmonic oscillator and so the dynamicalsymmetry group of the system is still SO (2 ,
1) with the transformed coordinates [35]. Theother important issue is that in this case the domain of the quantum particle is all of the realline and so it is not necessary to worry about the boundary condition at the origin. It is alsoeasy to extract the corresponding stochastic process by using the Hamiltonian as follows dU t = ( f e − U t + l ) dt + dB t , (5.21)where f = − ω and l = ω ( E − e/
2) + . It is also possible to extract the above equation byusing equation (5.11) and the Ito’s formula plus time re-parametrization. The connection ofthe Morse potential to functionals of the Brownian motion was discussed before in [36]. It iseasy to see by the Feynman-Kac formula that the Kernel of the Morse Potential is equal tothe following expectation in a stochastic process E [exp( λka t − λ A t ) | B t = y ]; (5.22)21ith λ = ω and k = E − e/ ω and a t = Z t exp( B s ) ds ; A t = Z t exp(2 B s ) ds. (5.23)In [36] the connection of the above process to the Maass Laplacian [37] were also discussedextensively. In this paper we explained many aspects of the connection between the Bessel process andits possible generalizations on the one side, and the conformal quantum mechanics and itsgeneralizations on the other side. The Bessel process as the path integral interpretationof conformal quantum mechanics has conformal symmetry before considering the non-Fellerboundary conditions of the process which correspond to the self adjoint extension of thecorresponding quantum mechanics. These boundary conditions could be Feller type if weconsider more generalized stochastic equations. We also discussed some generalizations ofthe Bessel process that have interesting well-known quantum system counterparts. Thesegeneralizations are based on the connection between the Green’s function of the quantumparticle and transition density of the Bessel process. Of course there are also many othersystems but we focused on those that have conformal symmetry as the dynamical symmetryof system. This work could be extended in many directions including a rigorous study of thestochastic processes that correspond to the self adjoint extension of the singular quantummechanics in the finite domain. This could be done by using the definition of local time.Investigating time inversion properties of the generalized Bessel processes in the wide sensecan be useful in classification of time invertible processes.Another interesting study could be the study of the process and quantum mechanics asa system with supersymmetry; in this case we will have superconformal symmetry as thesymmetry of the quantum mechanics.
Acknowledgments
I thank Benjamin Doyon, Shahin Rouhani and Roberto Tateo for careful reading of themanuscript and useful comments. I thank also Sebastian Guttenberg for stimulating discus-sions and reading the manuscript. 22
Appendix A: Self Adjoint Extension of the Hamiltonian
In this appendix we will summarize Von Neumann-Weyl method of self adjoint extensionfor the Hamiltonian operators [15, 38].Consider a Hilbert space H then an operator ( A, D ( A )) defined on H is said to be denselydefined if the subset D ( A ) is dense in H , i.e., that for any ψ ∈ H one can find in D ( A ) asequence φ n which converges in norm to ψ , in other words we should have R ∞ | ψ − φ n | dx < ǫ for arbitrary positive ǫ .The adjoint operator of an operator H with dense domain D ( H † ) is H † . The domain D ( H † ) is the space of functions ψ such that the linear form φ → ( ψ, Hφ ) is continuous forthe norm of H which guaranties the existence of a ψ † ∈ H such that( ψ, Hφ ) = ( ψ † , φ ) . (7.1)Then one may define H † ψ = ψ † . An operator ( H, D ( H )) is said to be symmetric orHermitian if for φ , ψ ∈ D ( H ) we have ( φ, Hψ ) = ( Hφ, ψ ). The operator H with the densedomain D ( H ) is said to be self-adjoint if D ( H † ) = D ( H ) and H † = H .Definition of the deficiency subspaces K ± are by K ± = { ψ ∈ D ( H † ) , H † ψ = ± iψ } , (7.2)with dimensions n ± which are called the deficiency indices of the operator H and will bedenoted by the ordered pair ( n + , n − ). The following theorem, discovered by Weyl and gen-eralized by Von Neumann, is the most important result of this appendix. Theorem : For an operator H with deficiency indices ( n + , n − ) there are three possibilities:1: If n + = n − = 0, then H is self-adjoint.2: If n + = n − = n , then H has infinitely many selfadjoint extensions, parameterized bya unitary n × n matrix with n real parameters.3: If n + = n − , then H has no self-adjoint extension.A relevant example for the above theorem is the Bessel operator discussed in the paper.The case on the half line was discussed extensively in section 2 and we will not discussit again but the case in the the finite interval [0 , L ] is more complicated and needs to bediscussed separately. In this case for ν ≥ , ≤ ν < , δ = 3 case for ν ≥
1. For the case 0 ≤ ν < L , as an infinite well with one parameter extension as the δ = 3 case. The boundary condition at the origin is just as before. The interesting point is,since for a differential operator of order n with deficiency indices ( n, n ), all of its self-adjointextensions have discrete spectrums one could argue that for this case all of the energy levelsare discrete. In the other words for the particle in the finite sphere with origin removed thespectrum of energy is completely discrete for 0 < δ < In this appendix we would like to summarize Feller’s classification of possible bound-ary conditions for the one dimensional stochastic equation [39, 40]. Consider the followingequation as our stochastic equation dx t = µ ( x t ) dt + σ ( x t ) dB t (8.1)To classify the possible boundary conditions we need to define the following two functions asthe scale function s ( x ) and speed measure m ( x ) as follows s ( x ) := exp( − Z x µ ( x ′ ) σ ( x ′ ) dx ′ ) , m ( x ) := 2 σ ( x ) s ( x ) . (8.2)Using the above functions one can define the following four different functions for diffusionin the interval with endpoints l and rS [ x, y ] = Z yx s ( z ) dz, S ( l, y ] = lim x → l + S [ x, y ] , S [ x, r ) = lim x → l − S [ x, y ] , (8.3) M ( c, d ) = Z dc m ( x ) dx, M ( l, y ] = lim x → l + M [ x, y ] , M [ x, r ) = lim x → l − M [ x, y ] , (8.4) X ( l ) = Z xl S ( l, y ] m ( z ) dz, X ( r ) = Z rx S [ z, r ) m ( z ) dz, (8.5) N ( l ) = Z xl S [ z, x ] m ( z ) dz, N ( r ) = Z rx S [ x, z ] m ( z ) dz. (8.6)24he boundary classification depends on the behavior of the above functions and one can putthe possible boundary conditions in one of the following four types for the endpoint e :1: regular if P ( e ) and N ( e ) be finite,2: exit if P ( e ) be finite and N ( e ) be infinity,3: entrance if P ( e ) be infinite and the N be finite,4: natural if P ( e ) and N ( e ) be infinity.For entrance, exit and natural, no boundary conditions are needed but for regular bound-ary, the conditional probability is not unique and dependent on the boundary conditions. Anexit boundary can be reached from the interior point of the domain with positive probabilityhowever it is not possible to start the process from the exit boundary. An entrance boundarycannot be reached from the interior point of the domain but it is possible to start the pro-cess from the entrance boundary. A natural boundary cannot be reached in finite time fromthe interior point of the domain and it is impossible to start the process from the naturalboundary. A regular boundary is accessible and could be reflecting if m ( e ) = 0 and stickyif m ( e ) >