Feynman Path Integral: rigorous formulation

Feynman Path Integral: Mathieu Beau, Postdoctoral researcher Dublin Institut for Advanced Studies (DIAS) Collaborator : Prof. Tony Dorlas, DIAS

Summary : 1. Feynman Path Integral : from a calculus to a rigorous definition 2. A rigorous Approach (Thomas-Bijma, Dorlas-Beau) : path distribution 3. et perspectives

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list 1. Feynman Path Integral : from a calculus to a rigorous definition

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Quantum mechanics and E.Schrödinger equation (1926) Ψ(x, t) L 2 (R d ), d > 0 satisfy the equation : ĤΨ(x, t) = i Ψ(x, t) t Ψ(x, 0) = ϕ(x) (1) where the intitial condition at t = 0 is known, and where the self 2 adjoint operator is given by Ĥ := 2m + V (x) The solution is given by : Ψ(x, t) = e ith ϕ(x) = + K(x, t; x 0, 0)ϕ(x 0 )dx 0 (2) where the propagator is solution of a Schrödinger equation : ĤK(x, t; x 0, 0) = i t K(x, t; x 0, 0) iδ(x x 0 )δ(t) (3)

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Path integral formulation, R.P.Feynman (1942) Action for a particle in an external potential V : S(x f, t f ; x i, t i ) = tf t i dt L(x(t), ẋ(t), t) = tf t i ( m dt 2 ẋ(t)2 V (x(t))). Then the solution given by : Ψ(x, t) = e ith ϕ(x) = + can be computed as a path integral : K(x, t; x 0, 0)ϕ(x 0 )dx 0 (4) K(x f, t f ; x i, t i ) = D[x(t)] e is(x f,t f ;x i,t i )/ (5) QUESTION : How can we define a Feynman path integral?

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Integral Calculus : We set m = 1 and = 1. Discretising the action to a finite subdivision σ = {t 1,..., t n } with 0 = t 0 < t 1 < < t n < T and (x 1,.., x n ) R n we can consider different boundary conditions. Here we consider the following BC : The discretised kinetic action is : S (K) σ (x, x n,.., 0) = 1 2 x(t = 0) = 0; x(t = T ) = x (6) ( (x xn ) 2 t t n + (x n x n 1 ) 2 t n t n 1 +... + x 2 1 t 1 )

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Slits Screen Source 2b x z 2a D L M. Beau, Feynman Path Integral approach to electron diffraction for one and two slits, analytical results, Eur. J. Phys. 33 (2012)

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list The corresponding Feynman distributions are as follows : ( ) exp is σ (K) (x, x n,.., 0) F σ (x, x n,.., 0), (7) (2iπ) n (t t n )(t n t n 1 )... t 1 The discretised potential action is : S (V ) σ (x, x n,.., 0) = (V (x)(t t n ) + V (x n )(t n t n 1 ) + + V (x 1 )t 1 ) The propagator is given by : K t (x, 0) lim dx 1... dx n F σ (x, x n,.., 0)e is (V ) σ (x,x n,..,0), (8) n R 1 R 1

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Examples : Free particle : K (0) (x, t; x 0, t 0 ) = 1 (2iπ (t t 0 )/m) Harmonic oscillator : (V (x) = mω2 2 x 2 ) ( K (ω) (x, t; x 0, t 0 ) = mω 2iπ sin(t ω) x x 0 2 2 (t t eim 0 ) d/2 ) d/2 e imω 4 ( x+x 0 2 tan( ωt 2 )+ x x 0 2 cotan( ωt 2 )) Notice that K (ω) (x, t; x 0, t 0 ) K (0) (x, t; x 0, t 0 ) when ω 0.

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list An Historical Overview 1923 : N.Wiener Brownian motion, Wiener measure 1926 : E.Schrodinger wave equation 1948 : R.Feynman article Lagrangian formulation of QM 1949 : M.Kac solution of heat equation as a path Integral 1960 : R.Cameron analytic continuation ( i m 1 ν + i m, νr1 ) 1967 : K.Itô Fresnel Integral on Hilbert space 1972 : C. De Witt-Morette Definition without limiting procedure 1976 : S. Albeverio and R. Høegh-Krohn Extension of Itô idea 1983 : T.Hida and L.Streit White noise analysis

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Principle interests 1976 : K. Itô, S. Albeverio and R. Høegh-Krohn Fresnel Integral on Hilbert space 1983 : T.Hida and L.Streit White noise analysis 2000 : E. Thomas Path distribution on sequence spaces

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list The Itô, Albeverio and Høegh-Krohn approach Let H be an Hilbert space with the inner product (.,.) and the norm.. Define F(H) as the space of bounded continuous fonction on H of the form : f (x) = e i(x,k) dµ(k) H for some µ M(H) (where M(H) is the Banach space of bounded complex Borel-measures on H). We define the normalized integral ( Fresnel integral ) on H by H e i γ 2 k 2 i 2 f (γ)dγ := e 2 dµ(k) (9) H

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Theorem (The Feynman-Itô formula) Let V and ϕ be Fourier transforms of bounded complex measure in R d. Let H be the real Hilbert space of continuous path γ from [0, t] to R d such that γ(t) = x and γ L 2 ([0, t]; R d ) with inner product (γ 1, γ 2 ) = t 0 γ 1(τ) γ 2 (τ)dτ, then the solution of the Schrödinger equation is given by : ψ(x, t) = H where γ 2 = t 0 γ(τ)2 dτ e i 2 γ 2 e i t 0 V (γ(τ))dτ ϕ(γ(0))dγ (10)

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list By the assumptions of the last Thm, we have V (x) = R d e iαx dµ(x) and ϕ(x) = R d e iαx dν(x), where µ and ν are in M(R d ). Then, by the proof of the Thm, they give an explicit formula to (10) : ψ(x, t) = ( i) n n=0 n! exp ( i 2 t 0 n j,l=0 where G t (t j, t l ) = t t j t l. t dt n dt 1 0 R 1 R 1 n G t (t j, t l )α j α l ) exp (ix α j )dν(α 0 ) j=0 n dµ(α j ) j=1

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list The Hida and Streit approach Idea : introduce a Gaussian measure dµ G (x) to define the integral as a product of duality : (2iπ) d/2 e i 2 x 2 f (x)dx = i d/2 e i+1 2 x 2 f (x)dµ G (x) R d R d Infinite dimensional : let the Hilbert space H = L 2 (R 1 ), the nuclear space E = S(R 1 ) and the corresponding dual space E = S (R 1 ). Let µ be the Gaussian measure on the Borel σ-algebra of E identified by the characteristic function : E e i X,ξ dµ(x ) = e 1 2 ξ 2 H

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list We denote by (L 2 ) = L 2 (E, µ) the Hilbert space with the inner product : φ, ϕ (L 2 ), (φ, ϕ) = E φ(x)ϕ(x)dµ(x). (L 2 ) is unitary equivalent to the symmetric Fock space F S (E C ), E C being the complexification of E. We introduce the Fourier-transform analogue φ (E ), (T φ)(ξ) := φ, e i X,ξ We have the following representation : for ϕ (L 2 ),!{F n L 2 S (Rn )} n 0 s.t. (T φ)(ξ) = n 1 2 0 i n e ξ 2 R n F n (t 1,, t n )ξ(t 1 ) ξ(t n ) ϕ 2 (L 2 ) = n 0 n! F n 2 0

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list A white noise distribution over E is defined as an element of the completion (E p ) of (L 2 ) w.r.t. the norm. p = H p. 0, where H := d2 dt 2 + 1 + u 2 (eigenvector : Hermite polynomial). One has the chain : (E) := p (E p ) (E 1 ) (L 2 ) (E 1 ) (E ) := p (E p ) Similary, we introduce the T -transform : φ (E ), (T φ)(ξ) := φ, e i X,ξ

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Theorem (Feynman-Hida-Streit formula) Let s define : F (ξ) = (2iπt) d/2 exp ( i 1 2 t t 0 ξ(s) 2 ) exp ( i 2t There exists a unique element φ (E ) such that Also we have : K(x, t; x 0, t 0 ) = T φ(ξ) = F (ξ). 1 (2iπt) d/2 e i 2 (x x 0) 2 = F (ξ = 0) ( t ) ) 2 x x 0 i ξ(s) t 0 where K(x, t; x 0, t 0 ) is the kernel of the operator e ith 0, H 0 = 2

Quantum mechanics and path integral Integral Calculus Rigorous Approaches : non-exhaustive list Advantages - disadvantages The Itô, Albeverio and Høegh-Krohn approach : (+) continuous path (-) small class of potential : V (x) = R d e iαx dµ(α), e.g. e x, (1 + x ) 2, some bounded and continuous potential. (+) recent results : x 4 step to QFT φ 4 The Hida and Streit approach : (+) wide class of potential : dν(x) = V (x)dx, e.g. V (x) = γδ(x x 0 ), γ R 1, x 0 R 1 V (x) = R d e αx dµ(α), e.g. γe ax, γ R 1, a R 1 (-) not defined for polynomial growing x δ, δ > 2 (-) paths are tempered distributions : not too much information on the nature of paths.

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion 2. A rigorous Approach (Thomas-Bijma, Dorlas-Beau) : path distribution

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Idea : For some potential V, we want to define the propagator as a scalar product between a path distribution F and e i t 0 V [x(t)] : K t (x, x 0 ) e i t 0 V [x(t)], F = De i t 0 V [x(t)], µ (11) Problems : (1) Does the limit n exist? What is µ and D? (2) F distribution? On which space of paths? (3) What is the meaning of,? (4) V belongs to a space of functions, which one is suitable? This is the final objective but we first want to look at the discrete-time analogue not easy to do and a lot to understand

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion As a first approach, we work on a discrete-time space, i.e. σ = {1, 2,.., n}, i.e. t j = j = 1, 2,..., n We first consider the following BC : The discretised kinetic action is : x(0) = 0; ẋ(n) = 0 (12) S σ (K) = 1 ( (xn x n 1 ) 2 +... + (x 2 x 1 ) 2 + x 2 ) 1 2 The corresponding Feynman distributions are as follows : F σ (x 1,.., x n ) = exp (is (K) σ ) (2iπ) n/2 (13)

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Definition The Feynman-Thomas measure µ σ on R n is defined by where µ σ (dx 1... dx n ) M (n) F σ (x 1,.., x n )dx 1... dx n (14) M (n) (x 1,.., x n ) = n j=1 0 ds j e s j /β j e x2 j /2s j, β j ]0, + [ R 1 β j 2πsj such that D (n) M (n) = δ (n), where D (n) ( ) n j=1 1 β j 2 2, and xj 2 hence, the Feynman-Thomas Distribution F σ on R n is given by F σ (x 1,.., x n )dx 1..dx n = D (n) µ σ (dx 1,.., dx n ), (15)

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Let the Fourier Transform of the distribution F σ be the following function : n F σ (ξ 1,.., ξ n ) = exp i x j ξ j F σ (x 1,.., x n )dx 1..dx n, (16) R n Explicit calculation gives F σ (ξ 1,.., ξ n ) = exp i 2 j=1 n K jl ξ j ξ l, where K jl = j l (17) j,l=1

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Hilbert space for paths Remark : Now a path is a sequence because the time is discrete. We introduce a family of Hilbert spaces of sequences labelled by a real parameter γ : l 2 γ = {(ξ j ) j=1 R j γ ξj 2 < + }. (18) j=1 This is a Hilbert space with inner product given by (ξ, ζ) γ = ξ j ζ j j γ j=1 Now we need to use the Sazonov Theorem to prove the existence of the projective limit µ = lim µ (n) on l 2 γ w.r.t. a weak topology.

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Theorem (Sazonov) Let (µ (n) ) n N 1 be a projective system of bounded measures on the dual H of a separable Hilbert space H. Assume that there exist positive measures ν n such that µ (n) ν n and which are uniformly bounded : sup n N 1 ν n < +, and such that the Fourier transforms Φ n : H C 1 given by Φ n (ξ) = e i ξ, x ν n (dx), are equicontinuous at ξ = 0 in the Sazonov topology. Then there exists a unique bounded Radon measure µ on H σ, where the subscript σ denotes the weak topology, such that π n(µ) = µ (n) for all n N 1.

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Idea of the proof : Equicontinuity in the Sazonov topology : for all ɛ > 0 there exists a Hilbert-Schmidt map u B(H) such that uξ 1 = Φ N (ξ) Φ N (0) ɛ n N 1. To determine the projective limit of the complex-valued measures µ (n) above, we apply this theorem to auxiliary positive measures which dominate µ (n). Step of the Proof : (1) Construction of an auxiliar measure ν n (such that µ (n) ν n ) (2) Conditions over γ to ensure the boundnedness of the measure (sup n ν n < + ) (3) Condition over γ to ensure the equicontinuity of ν n (4) Sasonov Theorem => Feynman-Thomas measure µ exists.

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Theorem Consider the map K : lγ 2 l γ 2 with K jl = j l, and assume γ > 7 2. Then there exists a unique path distribution F K on l γ 2 such that F K (ξ) = e i Kξ,ξ /2 given by F K = Dµ where D = j=1 ( 1 β j 2 2 x 2 j ) and where µ is a bounded Radon measure, strongly concentrated on l 2 γ w.r.t. the weak topology.

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Corollary Suppose that the potential V : R 1 R 1 belongs to E (2) (R), i.e. it is twice continuously differentiable with bounded first and second derivatives. Moreover, let (λ j ) j=1 be a sequence of positive constants such that j=1 β jλ j < +, where the constants β j satisfy the conditions of the above lemmas, in particular if β j = c i δ with δ > 5/2. Then the Feynman path integral exp i λ j V (x j ), F exists. j=1 Remark : In particular, we can take λ j = e ɛj for small ɛ > 0

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion This follows from the theorem since exp i λ j V (x j ), F = D exp i j=1 λ j V (x j ), µ where [ µ is the Feynman-Thomas measure. It therefore suffices if D exp i ] j=1 λ jv (x j ) is bounded. But D exp i = j=1 λ j V (x j ) = j=1 { 1 + 1 2 β j j=1 ( iλj V (x j ) + λ 2 j V (x j ) 2)} exp i λ j V (x j ). j=1

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Comment 1 : Discrete-time Schrödinger equation General boundary condition x k at t 0 = k for an arbitrary integer k. Formally, one then has F k = exp [ i 2 n=k+1 (x n x n 1 ) 2 ] n=k+1 ( ) dxn. 2iπ Denoting Ψ k playing the role of a wave function at time k : Ψ k (x k ) = exp i V (x j )λ j, F k, j=k There is then an obvious recursion relation : [ ] i Ψ k (x k ) = exp 2 (x k+1 x k ) 2 iv (x k+1 )λ k+1 Ψ k+1 (x k+1 ) dx k+1. 2iπ

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Comment 2 : Contact with Albeverio approach Assuming : V (x) = e ixy ν(dy) and Ψ k+1 (x) = e ixy µ k+1 (dy), we get the analogous Feynman-Itô formulae : Ψ k (x k ) = µ k (dy)e ixky, where f, µ k = ( i) n n=0 n! ν(dy 1 )... ν(dy n ) µ k+1 (dy)e i 2 y 2 f (y 1 + + y n + y) defines a bounded measure.

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Comment 3 : Scattering general boundary condition at T + : taking x 0 arbitrary, we define the classical path x i = x 0 + v i, where v = lim T + v T is the limiting velocity. Replacing x i by x i + x i in the MBC action it becomes S n = i n v 2 (x j + x j (x j 1 x j 1 )) 2 (T t n ) + 2 t j t j 1 = i 2 n j=1 j=1 (x j x j 1 ) 2 t j t j 1 + i 2 v 2 T + iv(x n x 0 ).

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion One defines the kernel of the (adjoint) wave operator (Ω ) at momentum k out = v (remember that = 1 and m = 1 so that v = k m = k) by omitting these factors and then taking n. In the discrete-time case we obtain (Ω ) (k out, x 0 ) = exp i V (x j + x 0 + k out j)λ j ik out x 0, F. j=1 The scattering matrix is defined by S(k out, k in ) = (Ω ) (k out, x 0 )Ω + (k in, x 0 ) In this case of course we must take λ j = e ɛ j. If V decays sufficiently fast for x +, it is known that the limit ɛ 0 exists.

Program Feynman-Thomas measure and distribution on R n Feynman-Thomas measure and distribution on l 2 γ Results - discussion Comment 4 : Continuous-time For boundary conditions x(0) = 0, x(t) = x we have : x(τ) = x c (τ) + a n φ n (t), where φ(t) = 2/t sin πτ/t and x c (τ) = (x/t)τ (for e.g.). So we get : 1 t dτẋ(τ) 2 = ix 2 2 0 2t + i π 2 n 2 2t 2 a2 n n=1 Method : construct an auxiliar F-T measure on a sequence space, apply the discrete time approach and come back to the path space. we showed that the F-T measure exists and is strongly concentrated on L 2 ([0, t]) : Not satisfactory. j=1

3. and future projects

Main Conclusion : (i) Different approaches with their advantages and inconvenients : class of potential, path spaces, time-dependant potential (ii) Question of path spaces, continuous? Future Projects (i) Extension of Thomas approach for V = ax 2 : similar work, modifying the expression of F (ii) Contruct a F-T measure for continuous-time F-T measure strongly concentrated on C 0 ([0, t]) (Prokhorov Theorem). (iii) For some potential reasonably nice (bounded and continuous), singular (Delta), quartic (x 4 )

Interests (I) Wide applications : scattering theory, diffraction theory, magnetic field, semi-classical approximation, time dependent Gibbs states, QFT (II) Statistical Mechanics : Feynman-Kac Integral (fermion, boson, polymers) (III) Pure Mathematics : Infinite dimensional analysis, EDP.

References - S. A. Albeverio and R. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals. Springer Lecture Notes in Mathematics 523, 1976. - T. Hida et L.Streit, White Noise : An infinite dimensional calculus. Kluwer, Dordrecht (1995) - E. Thomas, Path distributions on sequence spaces. Proc. Conf. on Infinite-dimensional Stoch. Anal. Neth. Acad. Sciences, 1999, 235 268. - M. Beau, T. Dorlas, Discrete-Time Path Distributions on Hilbert Space, Indagationes Mathematicae 24, 212-228 (2012) [arxiv :1202.2033]