Extending coherent state transforms to Clifford analysis William D. Kirwin, Jos´e Moura˜o, Jo˜ao P. Nunes and Tao Qian ∗ † † ‡ Abstract 6 1 0 Segal-Bargmann coherent state transforms can be viewed as unitary maps from L2 2 spacesoffunctions(orsectionsofanappropriatelinebundle)onamanifoldX tospaces v ofsquareintegrableholomorphicfunctions(orsections)onXC. Itisnaturaltoconsider o higher dimensional extensions of X based on Clifford algebras as they could be useful N in studying quantum systems with internal, discrete, degrees of freedom corresponding 7 to nonzero spins. Notice that extensions of X based on the Grassman algebra appear naturally in the study of supersymmetric quantum mechanics. In Clifford analysis the ] h zero mass Dirac equation provides a natural generalization of the Cauchy-Riemann p conditions of complex analysis and leads to monogenic functions. - h For the simplest but already quite interesting case of X = R we introduce two t a extensions of the Segal-Bargmann coherent state transform from L2(R,dx) R to m m ⊗ Hilbert spaces of slice monogenic and axial monogenic functions and study their prop- [ erties. These two transforms are related by the dual Radon transform. Representation 3 theoretic and quantum mechanical aspects of the new representations are studied. v 0 8 Contents 3 1 0 1 Introduction 2 . 1 0 6 2 Preliminaries 2 1 2.1 Coherent state transforms (CST) . . . . . . . . . . . . . . . . . . . . . . . . 2 : v 2.2 Clifford analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 i X 3 Monogenic extensions of analytic functions 5 r a 3.1 Slice monogenic extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Axial monogenic extension and dual Radon transform . . . . . . . . . . . . . 5 4 Clifford extensions of the CST 8 4.1 Slice monogenic coherent state transform (SCST) . . . . . . . . . . . . . . . 8 4.2 Axial monogenic coherent state transform (ACST) . . . . . . . . . . . . . . . 10 5 Representation theoretic and quantum mechanical interpretation 11 ∗Mathematics Institute, University of Cologne †Department ofMathematics andCenterfor MathematicalAnalysis,GeometryandDynamicalSystems, Instituto Superior T´ecnico, University of Lisbon ‡Department of Mathematics, Faculty of Science and Technology, University of Macau 1 1 Introduction Clifford analysis (see [BDS, DSS]) has been developed to extend the complex analysis of holomorphic functions to Clifford algebra valued functions, satisfying properties generalizing the Cauchy–Riemann conditions. On the other hand, in quantum physics, Clifford algebra or spinor representation valued functions describe some systems with internal degrees of freedom, such asparticles with spin. Recall that the Segal-Bargmann transform [Ba, Se1, Se2], for a particle on R, establishes the unitary equivalence of the Schro¨dinger representation with Hilbert space L2(R,dx), with (Fock space-like) representations with Hilbert spaces, L2(C,dν), of holomorphic functions H on the phase space, R2 = C which are L2 with respect to a measure ν. In the Schro¨dinger ∼ representation the position operator xˆ acts diagonally while the momentum operator is Sch pˆ = i d . On the other hand, on the Segal–Bargmann Hilbert space L2(C,dν) it is the Sch dx H operator xˆ +ipˆ that acts as multiplication by the holomorphic function x+ip. SB SB In [Ha1], Hall has defined coherent state transforms (CSTs) for compact Lie groups G which are analogs of the Segal-Bargmann transform. These CSTs correspond to applying heat kernel evolution, e∆2, followed by analytic continuation to the complexification GC of G [Ha2]. We use the fact that, after applying the heat kernel evolution, the resulting functions are in fact extendable to Rm+1 in two natural ways motivated by Clifford analysis. These will lead to two generalizations of the CST, the slice monogenic CST, U , and the axial s monogenic CST, U , which take values on spaces of C –valued functions on Rm+1, where a m C denotes the complex Clifford algebra with m generators. One, = ImU , is a subspace m s s H of the recently introduced space of square integrable slice monogenic functions [CoSaSt1], while the other, = ImU , is a Hilbert space of, the more traditional in Clifford analysis, a a H axial monogenic functions [BDS, DSS]. We show that the two coherent state transforms are related by the dual Radon transform Rˇ, U = Rˇ U . a s ◦ A possibly interesting alternative way of defining a monogenic CST would be through Fueter’s theorem [F, Q, KQS, PQS, Sc]. It would be very interesting to relate such a transform with the one studied in the present paper. As in the case of the usual CST, the aim of these transforms is to describe the quantum states of a particle in R with internal degrees of freedom parametrized by a Clifford algebra, through slice/axial monogenic functions, thus making available, the powerful analytic ma- chinery of Clifford analysis. In Section 5, we show that the operator xˆ +ipˆ has a simple 0 0 action in both the slice and axial monogenic representations. 2 Preliminaries 2.1 Coherent state transforms (CST) Let G be a compact Lie group with complexification GC. In 1994, Hall [Ha1] introduced a class of unitary integral transforms on L2(G,dx), where dx is Haar measure, to spaces of holomorphic functions on GC which are L2 with respect to an appropriate measure. These 2 are known as coherent state transforms (CSTs) or generalized Segal–Bargmann transforms. These transformswere extended togroupsofcompact type, whichinclude thecaseofG = Rn considered in the present paper, by Driver in [Dr]. General Lie groups of compact type are direct products of compact Lie groups and Rn, see Corollary 2.2 of [Dr]. For G = Rn these transforms coincide with the classical Segal–Bargmann transform [Ba, Se1, Se2]. We will briefly recall now the case G = R, which we will extend to the context of Clifford analysisinthepresent paper. Thecaseofarbitrarygroupsofcompacttypeisveryinteresting and will be studied in a forthcoming work. Let ρ (x) denote the fundamental solution of the t heat equation. ∂ 1 ρ = ∆ρ , ∂t t 2 t i.e. ρt(x) = (2π1t)1/2 e−x22t, where ∆ is the Laplacian for the Euclidian metric and let (C) denote the space of entire H holomorphic functions on C. The Segal–Bargman or coherent state transform U : L2(R,dx) (C) −→ H is defined by U(f)(z) = ρ (z x)f(x)dx = 1 − R Z = 1 e−(z−2x)2 f(x)dx. (2.1) (2π)1/2 R Z whereρ hasbeenanalyticallycontinuedtoC. WeseethatthetransformU in(2.1)factorizes 1 according to the following diagram (C) )(cid:9)♠♠♠♠♠U♠♠♠♠♠♠♠♠66 H OOC (2.2) L2(R,dx)(cid:31)(cid:127) // (R) ∆ A e2 where (R)isthespaceof(complexvalued)realanalyticfunctionsonRwithuniqueanalytic A continuation to entire functions on C, denotes the analytic continuation from R to C and e∆(f) is the (real analytic) heat kernelCevolution of the function f L2(R,dx) at time t = 1, 2 ∈ that is the solution of ∂ h = 1∆h ∂t t 2 t , (2.3) h = f 0 (cid:26) evaluated at time t = 1, ∆ e2(f) = h1. Let (R) (R) denote the image of L2(R,dx) by the operator e∆. 2 A ⊂ A e 3 Theorem 2.1 (Segal–Bargmann) The transform (2.1) L2(C,νdxdy) ❥❥❥❥❥❥❥U❥❥❥❥❥❥H❥❥❥55 OOC (2.4) L2(R,dx) // (R) ∆ A e2 e is a unitary isomorphism, where z = x+iy C,x,y R and ν(y) = e−y2. ∈ ∈ 2.2 Clifford analysis Clifford analysis has beendeveloped to extend thecomplex analysis ofholomorphic functions to Clifford algebra valued functions, satisfying properties generalizing the Cauchy–Riemann conditions [BDS, DSS]. Let us briefly recall from [CoSaSt1] and [DS], some definitions and results from Clifford analysis. Let R denote the real Clifford algebra with m generators, m e ,j = 1,...,m, identified with the canonical basis of Rm R and satisfying the relations j m ⊂ e e +e e = 2δ . We have that R = m Rk , where Rk denotes the space of k-vectors, i j j i − ij m ⊕k=0 m m defined by R0 = R and Rk = span e : A 1,...,m , A = k . We see that, in m m R{ A ⊂ { } | | } particular, Rm is identified with the space of 1-vectors, Rm = R1 , x = m x e and Rm+1 m j=1 j j is identified with the space, R≤1, of paravectors of the form, m P m ~x = x +x = x + x e . 0 0 j j j=1 X Notice also that R = C and R = H. The inner product in R is defined by 1 ∼ 2 ∼ m < u,v >=< u e , v e >= u v , A A B B A A A B A X X X and therefore, x2 = x 2 = < x,x > . The Dirac operator is defined as −| | − m ∂ = ∂ e , x xj j j=1 X and the Cauchy-Riemann operator as ∂ = ∂ +∂ . ~x x0 x We have that ∂2 = m ∂2 and ∂ ∂ = m ∂2 . x − j=1 ∂x2j ~x ~x j=0 ∂x2j Recall that a continuously differentiable function f on an open domain U Rm+1, P P ⊂ with values on R or C = R C, is called (left) monogenic on U if (see, for example, m m m ⊗ [BDS, DSS]) ∂ f(x ,x) = (∂ +∂ )f(x ,x) = 0. ~x 0 x0 x 0 For m = 1, monogenic functions on R2 correspond to holomorphic functions of the complex variable x +e x . 0 1 1 4 3 Monogenic extensions of analytic functions 3.1 Slice monogenic extension Recall from [CoSaSt1, CoSaSt2] that a function f : U Rm+1 R is slice monogenic if, m ⊆ → for any unit vector ω Sm−1 = x R1 : x = 1 , the restrictions f of f to the complex ∈ { ∈ m | | } ω planes H = u+vω, u,v R , ω { ∈ } are holomorphic, (∂ +ω∂ )f (u,v) = 0, ω Sm−1. (3.1) u v ω ∀ ∈ Let (Rm+1) denote the space of slice monogenic functions on Rm+1. From the definition SM of (R) in diagram (2.2) and the Remark 3.4 of [CoSaSo] (see also Proposition 2.7 in A [CoSaSt3]) one obtains the following. Theorem 3.1 The slice-monogenic extension map, M : (R) R (Rm+1) s m A ⊗ −→ SM M (h)(x ,x) = M ( h e )(x ,x) = s 0 s A A 0 A X = hA(x0 +x)eA := exdxd0 hA(x0)eA = (3.2) A A X X ∞ xk dkh = A(x )e , k! dxk 0 A A k=0 0 XX is well defined and satisfies M (h)(x ,0) = h(x ), x R. s 0 0 0 ∀ ∈ 3.2 Axial monogenic extension and dual Radon transform A monogenic function f(x ,x) is called axial monogenic (see [DS], p. 322, for the definition 0 of axial monogenic functions of degree k) if it is of the form f(x ,x) = f (x ,x)e 0 A 0 A A X x f (x ,x) = B (x , x )+ C (x , x ), (3.3) A 0 A 0 | | x A 0 | | | | where B ,C are scalar functions and the functions f are monogenic. The monogeneicity A A A condition, ∂ f = ∂ f + ∂ f = 0, then leads to the Vekua–type system for B ,C , ~x A x0 A x A A A generalising the Cauchy-Riemann conditions, m 1 ∂ B ∂ C = − C , ∂ C +∂ B = 0, r = x . x0 A − r A r A x0 A r A | | Let (Rm+1) denote the space of axial monogenic functions on Rm+1. AM Axial monogenic functions are determined by their restriction to the real axis, f(x ,0). 0 The inverse map of extending (when such an extension exists) a real analytic function h on 5 R to an axial monogenic function on Rm+1 is called generalized axial Cauchy-Kowalewski extension and has been studied by many authors (see, for example, [DS]). Using the dual Radon transform to map slice monogenic functions to monogenic func- tions as proposed in [CLSS], we will factorize the axial monogenic extension into the slice monogenic extension followed by the dual Radon transform. Let us first recall the definition of the dual Radon transform. (See, for example, [He].) Definition 3.2 The dual Radon transform of a smooth function f on Rm+1 is Rˇ(f)(x ,x) = f(x ,< x,t > t)dt. (3.4) 0 0 ZSm−1 It is known from [CLSS] that Rˇ maps entire slice monogenic functions to entire monogenic functions. Let us denote a function f (R) and its analytic continuation to the complex plane ∈ A H by the same symbol, f. The following is a small modification of the Theorem 4.2 in [DS]. t Theorem 3.3 The axial monogenic or axial Cauchy-Kovalewski extension map M : (R) R (Rm+1) a m A ⊗ −→ AM M (h)(x ,x) = M ( h e )(x ,x) = a 0 a A A 0 A X = h (x + < x,t > t)dte , (3.5) A 0 A A ZSm−1 X where dt denotes the invariant normalized (probability) measure on Sm−1, is well defined and satisfies M (h)(x ,0) = h(x ), x R = R0 . a 0 0 ∀ 0 ∈ m Proof. From (3.2) and (3.4) we see that the map M in (3.5) factorizes to a M = Rˇ M . (3.6) a s ◦ The fact that the image of this map is a subspace of the space of entire monogenic functions on Rm+1 is a consequence of the theorem A of [CLSS]. We still need to show that the functions M (h) are axial monogenic for all h (R) R . Notice that the Taylor series a m ∈ A ⊗ of h, with center at any point of R has infinite radius of convergence. Using (3.2), Theorem 3.1, and the fact that for ω Sm−1 one has ω2k = ( 1)k, we obtain ∈ − M (h)(x ,x) = M ( h e )(x ,x) = a 0 a A A 0 A X ∞ ( x,ω ω)k = Rˇ M (h )(x ,x)e = h i h(k)(x )dωe ◦ s A 0 A k! A 0 A A A ZSm−1 k=0 X X X ∞ ( 1)j ( 1)j = − h(2j)(x ) x,ω 2j +ω − h(2j+1)(x ) x,ω 2j+1dω e . (2j)! A 0 h i (2j +1)! A 0 h i A A j=0 ZSm−1 (cid:19) X X 6 and therefore, ∞ ( 1)j ( 1)j M (h)(x ,x) = − h(2j)(x )C x 2j +x − h(2j+1)(x )C x 2j e , a 0 (2j)! A 0 m,2j| | (2j +1)! A 0 m,2j+2| | A ! A j=0 X X where π C = sinm−1(θ)cos2j(θ)dθ. m,2j Z0 This is of the form (3.3) which completes the proof. We therefore get the following commutative diagram. (Rm+1) ❥❥❥❥❥❥M❥❥a❥❥❥❥❥❥A❥55 M OO A(R)⊗Rm❚❚❚❚❚❚M❚❚s❚❚❚❚❚❚❚)) Rˇ (3.7) (Rm+1) SM As an illustration let us consider the axial monogenic extension of plane waves ϕ , with p ϕ (x ) = eipx0. The axial monogenic extension of ϕ follows from Example 2.2.1 and Remark p 0 p 2.1 of [DS], where the axial monogenic extension of ex0 is given in terms of Bessel functions, by taking k = 0 and replacing x by ipx in the expressions of Example 2.2.1 of [DS]. Proposition 3.4 The axial monogenic plane waves are given by m 2i m/2−1 x M (ϕ )(x ,x) = Γ( ) I (p x )+i I (p x ) eipx0, (3.8) a p 0 2 p x m/2−1 | | x m/2 | | (cid:18) | |(cid:19) (cid:18) | | (cid:19) where I are the hyperbolic Bessel functions. α Proof. By representing, as in example 2.2.1 of [DS], M (ϕ )(x ) in the form a p 0 ∞ M (ϕ )(x ,x) = c xjB eipx0, a p 0 j j j=0 X and expressing the monogeneicity of the transform ∞ (∂ +∂ ) c xjB eipx0 = 0, x0 x j j j=0 X we obtain the following recurrence relation for the functions B (x ), j 0 B (x ) = ipB (x ) B′(x ), B (x ) = 1. j+1 0 − j 0 − j 0 0 0 The solution is B (x ) = ( ip)j. Then we see that the transform is obtained by replacing x j 0 − by ipx in the expressions of Example 2.2.1 of [DS]. 7 Remark 3.5 From Theorem A of [CLSS], Rˇ : (Rm+1) (Rm+1) is an injective SM → AM map. In fact, from Corollary 4.4 of [CLSS], we see that (non-zero) slice monogenic functions do not belong to KerRˇ. ♦ Remark 3.6 Note that, as in [DS], considering h (R) C , one also has, m ∈ A ⊗ M (h)(x ,x) = h (x +i x,t )(1 it))dte , (3.9) a 0 A 0 A h i − A ZSm−1 X which is equivalent to (3.5) and can be readily verified by expansion in power series. ♦ 4 Clifford extensions of the CST The two extensions (3.2) and (3.5) give two natural paths to define coherent state transforms by replacing the vertical arrow of analytic continuation in the diagram (2.4). Wereferthereaderinterestedintherepresentationtheoreticandthequantummechanical meaning of the Hilbert spaces introduced in the present section to section 5. 4.1 Slice monogenic coherent state transform (SCST) The slice monogenic CST is naturally defined by substituting the vertical arrow in the diagram (2.4) by the slice monogenic extension (3.2) leading to (Rm+1) C '(cid:7)✐✐✐✐✐✐✐✐U✐s✐✐✐✐✐✐✐S✐✐M44 OO Ms ⊗ (4.1) L2(R,dx ) C (cid:31)(cid:127) // (R) C 0 m m ⊗ ∆0 A ⊗ e 2 e where ∆ = d2 . Notice that even though the plane waves, ϕ (x ) = eipx0, are not in the 0 dx2 p 0 0 Hilbert space L2(R,dx ), they are generalized eigenfunctions of ∆ with eigenvalue p2, and 0 0 − therefore e∆20(ϕp)(x0) = e∆20 eipx0 = e−p22 eipx0 = e−p22ϕp(x0). (4.2) On the other hand since the plane waves ϕ (R) we can use (3.2) to obtain the following p ∈ A very simple result. Lemma 4.1 The slice monogenic plane waves are given by x M (ϕ )(x ) = M (eipx0) = eip~x = cosh(p x )+i sinh(p x ) eipx0. (4.3) s p 0 s | | x | | (cid:18) | | (cid:19) Proof. From (3.2) we obtain ∞ (ipx)k x M (ϕ )(x ) = eipx0 = cosh(p x )+i sinh(p x ) eipx0. s p 0 k! | | x | | k=0 (cid:18) | | (cid:19) X 8 Proposition 4.2 Let f L2(R,dx ) and 0 ∈ 1 f(x ) = eipx0f˜(p)dp. 0 √2π R Z We have Us(f)(x0,x) = 1 e−p22 eip~xf˜(p)dp = (4.4) √2π R Z = 1 e−p22 eipx0 cosh(p x )f˜(p)dp+i x 1 e−p22 eipx0 sinh(p x )f˜(p)dp √2π R | | x √2π R | | Z | | Z Proof. This result follows from Lemma 4.1, (3.2) and (4.2). Consider the standard inner product on C . Our main result in this section is the m following. Theorem 4.3 The SCST, U in Diagram (4.1), is unitary onto its image for the measure s dν on Rm+1 given by m 2 1 e−|x|2 dν = dx dx, m √π Vol(Sm−1) x m−1 0 | | where Vol(Sm−1) denotes the volume of the unit radius sphere in Rm, i.e. the map U in the s diagram ✐✐✐✐✐✐✐U✐s✐✐✐✐✐✐✐✐✐✐✐✐44HOOsMs (4.5) L2(R,dx ) C // (R) C 0 m m ⊗ ∆0 A ⊗ e 2 is a unitary isomorphism, where = U (L2(R,dx ) eC ) L2(Rm+1,dν ). s s 0 m m H ⊗ ⊂ SM Proof. Let (R) be the space of Schwarz functions on R. For f,h (R) R , with m S ∈ S ⊗ f = f e ,h = h e we have A A A A A A P P 2 1 e−|x|2 < U (f),U (h) > = e2ixp e−p2f˜ (p)h˜ (p) dmxdp = s s √πVol(Sm−1) 0 A A x m−1 A ZR×Rm | | X (cid:2) (cid:3) 2 1 e−|x|2 = e−p2f˜ (p)h˜ (p) cosh(2 x p) dmx dp = √πVol(Sm−1) A A | | x m−1 R Rm ! A Z Z | | X 2 1 ∞ = e−p2f˜ (p)h˜ (p) cosh(2up)e−u2du dp = √πVol(Sm−1) A A A ZR (cid:18)Z0 (cid:19) X = f˜ (p)h˜ (p)dp =< f,h > . A A R A Z X From the density of (R) C in L2(R) we conclude that U is unitary onto its image. s S ⊗ Remark 4.4 For each complex plane H := u+vω : u,v R and for f L2(R,dx) C , ω m { ∈ } ∈ ⊗ f = f e , the map f U (f) coincides, for each component f of f, with the A A A 7→ s Hω A Segal–Bargmann transform, which is surjective to L2(H ,dν) and unitary for the measure P (cid:12) H ω dν = e−v2dudv on H . (cid:12) ♦ ω 9 4.2 Axial monogenic coherent state transform (ACST) The axial monogenic CST is also naturally defined as the heat kernel evolution followed by the axial Cauchy-Kowalewski extension Ua = Ma e∆20 , ◦ i.e. by substituting the vertical arrow in the diagram (2.2) by the axial monogenic extension (3.5) (Rm+1) C &(cid:6)❤❤❤❤❤❤❤U❤a❤❤❤❤❤❤❤A❤❤M44 OO Ma ⊗ (4.6) L2(R,dx ) C (cid:31)(cid:127) // (R) C 0 m m ⊗ ∆0 A ⊗ e 2 The following is an easy consequence of Theorem 4.3, (3.6) and Remark 3.5. Theorem 4.5 Let (Rm+1) C denote the image of L2(R,dx ) C under U . a 0 m a H ⊂ AM ⊗ ⊗ The restriction of the dual Radon transform to defines an isomorphism to . s a H H The diagram ❡❡❡❡❡❡❡❡❡❡❡❡❡❡U❡a❡❡❡❡❡❡❡❡❡♠❡♠❡♠❡♠❡♠M♠❡♠❡a♠❡♠❡♠❡♠❡♠❡♠❡♠2266 HOOa L2(R,dx0)⊗Cm ❨❨❨❨e❨∆❨0❨/2❨❨❨❨❨// A❨e❨U(❨sR❨❨)❨❨⊗❨❨C❨❨◗m❨◗❨◗❨◗◗❨◗❨M◗❨◗❨s◗❨◗❨◗❨◗❨◗❨◗❨◗((,, Rˇ. (4.7) s H is commutative and its exterior arrows are unitary isomorphisms for the inner product on given by , , Ha h· ·iHa F,G = (Rˇ)−1(F)(Rˇ)−1(G)dν , (4.8) h iHa m Rm+1 Z where dν was defined in Theorem 4.3. m Proof. The injectivity of Rˇ follows from Remark 3.5. From (3.6), we conclude that |Hs U = Rˇ U which implies the surjectivity of Rˇ : . Then, the inner product a ◦ s |Hs Hs −→ Ha (4.8) is well defined, the diagram (4.7) is commutative and the exterior arrows are unitary isomorphisms. Remark 4.6 As mentioned in the introduction, a possibly interesting alternative way of defining a monogenic CST would be by replacing the dual Radon transform in (3.7) and in diagram(4.7)bytheFuetermapping,∆m−1,where∆ = m ∂2 (see[F,Q,KQS,PQS,Sc]). 2 j=0 ∂x2 j Notice however that the map ∆m2−1 Ms does not corrPespond to a monogenic extension of ◦ analytic functions of one variable as the restriction to the real line does not give back the original functions. It leads nevertheless to an interesting transform and it would be very interesting to relate it with U . a ♦ 10