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of p-adic Heisenberg group. For the Schrödinger representation, we can define Weyl operator and its kernel function, while for the induced representations from the characters of maximal abelian subgroups of Heisenberg group generated by the isotropic subspaces or self- in the p-adic symplectic , we calculate the Maslov index defined via the intertwining operators corresponding to the representation transformation operators in quantum mechanics.

2. BRIEF REVIEW OF p-ADIC ANALYSIS k,vvz,ak The details of p-adic analysis can be found in [8, 9, 10]. We only recall some basic materials used in this paper. Let p be a prime positive number, the field Qp of p-adic numbers is the completion of the rational number field Q with respect to the p-adic norm . Any nonzero element x Q can be written uniquely as x = pk x pn where | · |p ∈ p n≥0 n x 0, ,p 1 , x = 0 and k Z, and the p-adic norm of x is given by x = 1 . The subset Z of Q defined n ∈ { ··· − } 0 6 ∈ | |p pk P p p as Z = x Q : x 1 = x = x pn : x 0, ,p 1 = lim Z/pnZ is an integral domain called p { ∈ p | |p ≤ } { n≥0 n n ∈ { ··· − }} n the ring of p-adic integers. The ring Zp is a principal ideal domain, more precisely, its←− ideals are the principal ideals 0 and k P { } p Zp(k N). In particular, p = pZp is the unique maximal ideal of Zp, the corresponding residue field is Fp = Zp/pZp. ∈ 1 × Qp is also defined as the fraction field of Zp, i.e. Qp = Zp[ p ]. The group multiplicative Zp of invertible elements in the ring Z consists of the p-adic integers with unit norm, namely Z× = x Z : x = 1 = x = x pn : x p p { ∈ p | |p } { n≥0 n n ∈ 0, ,p 1 , x = 0 , hence the subset consisting of nonzero p-adic number is given by Q× = pmZ×. For any { ··· − } 0 6 } p mP∈Z p p-adic number x = x pn+k one defines the fractional part of x as [x] = −k−1 x pn+k Z[ 1 ], and for any two n≥0 n n=0 n ∈` p p-adic numbers x and y, the difference [x + y] [x] [y] Z Z[ 1 ] = Z. Call B(a; n) = x Q : x a p−n P − − ∈ p ∩ p P { ∈ p | − |p ≤ } a p-adic ball with a. Note that every point in B(a; n) is a center. The p-adic balls are both open and closed sets since B(a; n) = x Q : x a < p−n+1 , and they are disconnected sets since B(a; n) = p−1 B(a + pnx; n + 1). { ∈ p | − |p } x=0 Moreover one can show that Qp = nB(a; n) is a locally compact and totally disconnected topological field. An additive character χ on Q is∪ a homomorphism χ : Q C∗ with the property χ(x + y`) = χ(x)χ(y). The group p p → of additive characters of the field Qp is isomorphic to its Qp, where the is given by the mapping u χ(ux) = χu(x) := exp(2πi[ux]). Being locally compact, Qp has a real-valued Haar measure, i.e., a translation invariant7→ measure dx with the property d(x + a) = dx. If the Haar measure is normalized so that for the compact subring × n Zp it satisfies dx = 1, then dx is unique. For any a Q we ave d(ax) = a pdx. Let Q denote the space of product Zp ∈ p | | p of n-copies of Q . The p-adic norm of x = (x , , x ) Qn is given by x = max x , which is also a non- R p 1 n p p 1≤j≤n j p ··· ∈ | | | | n 1 n Archimedean norm. The Haar measure dx on Qp can be extended to a translation invariant measure d x = dx dx on n n n n n n ···n n Qp in the standard way, which has the properties: d (x + a)= d x (a Qp ), d (Ax)= det A pd x where A : Qp Qp ∈ n | α | → is a linear isomorphism such that det A = 0. Let K be a measurable subset in Qp , and L (K) (α 1) be a set of all 6 α n 1 n ≥ measurable functions f : K C, such that K f(x) d x < . A function f L (Qp ) is called integrable if there → n | | ∞ ∈ n exists limN→∞ n f(x)d x. We call this limit an improper integral and denote it as Qn f(x)d x. In particular, (B(0;−N)) R p ∞ f(x)dx = ν f(x)dx. We present a general formula of change of variables in integrals. If x(y) is an Qp R ν=−∞ |x|p=p R analytic of a clopen (close and open) set K Q onto a clopen set K Q , and x′(y) =0,y K , then R P R 1 p p 1 for any 1 we have ′ ⊂ . ⊂ 6 ∈ f L (K) K f(x)dx = K1 f(x(y)) x (y) pdy The following∈ integrals will be very useful. | | R R ν 1 ν dx = p (1 ). • |x|p=p − p ν 1 −ν ; R p (1 p ), ξ p p ν−−1 | | ≤ −ν+1 ν χ(ξx)dx = ; |x|p=p p , ξ p = p •  − | | −ν+2  0, ξ p p . R 1 1 | | ≥−ν 1 1 p f( x )χ(ξx)dx = (1 ) p f( ν ) f( ), ξ =0. Qp p p |ξ|p ν≥0 p |ξ|p |ξ|p |ξ|p • | |  − 2 − 6 2 λp(a) b k RQ χ(ax + bx)dx = χ( 4aP),p =2, where for a non-zero p-adic number a = p (a0 + a1p + ), λp(a)= • p √|a|p − 6 ··· R 1, k is even; ( a0 ) k is odd and k 1 (mod4); a0 p with ( p ) denoteing the Legendre symbol.  a0 ≡  i( p ), k is odd and k 3 (mod4), ≡ 2 2 λ2(a) b , where for a non-zero 2-adic number k 2 , Q2 χ(ax + bx)dx = χ( ) a =2 (1+2a1 +2 a2 + ) λ2(a)= •  √|a|2 − 4a ··· R 1 + ( 1)a1 i, k is even; ( 1)a−1+a2 (1 + ( 1)a1 i) k is odd .  − − 2 n n 2 n 2 The set L (Qp ) is the with the scalar product (f,g)= Qn f(x)¯g(x)d x for f,g L (Qp ) so that f L = p ∈ || || 2 n (f,f). In L (Q ) the Cauchy-Bunjakovsky inequality holds: (f,g) f 2 g 2 . The Fourier transform f p | R | ≤ || ||L · || ||L → p SYMPLECTIC GROUP AND HEISENBERG GROUP IN p-ADICQUANTUMMECHANICS 3

2 n 2 n ˆ n −1 ˆ F [f] maps L (Qp ) onto L (Qp ) one-to-one, where F [f](ξ) = f(ξ) = Qn f(x)χ(ξ x)d x and F [f](x) = f(x) = p · ˆ n Qn f(ξ)χ( ξ x)d ξ. Moreover, the Parseval-Steklov equality holds: (f,gR ) = (F [f], F [g]), i.e. the Fourier transform is a p − · unitary operator in L2(Qn). A complex-valued function ψ defined over Qn is called locally constant if for any x Qn there R p p ∈ p exists an integer l(x) Z such that ψ(x + x′)= ψ(x) when x′ (B(0; l(x)))n, and l : Qn Z is called a ∈ ∈ − p → function associated with ψ. In particular, characteristic functions themselves are locally constant. We denote by (Qn) the E p space of locally constant functions over Qn, by (Qn) the space consisting of all compactly supported functions belong to p D p (Qn), and by ∗(Qn) the set of all linear functionals on (Qn). Then (Qn) is dense in L2(Qn). If ψ (Qn), we have E p D p D p D p p ∈ D p constant characteristic functions, i.e. there exists l Z such that ψ(x + x′)= ψ(x) for any x Qn, x′ (B(0; l))n. ∈ ∈ p ∈ − 3. p-ADIC SYMPLECTIC GROUP

Consider a 2n-dimensional vector space (classical ) V over Qp, endowed with a non-degenerate alternat- ing J. Let Gr(n, V ) be the Grassmannian variety of n-dimensional subspaces of V , and Lag(n, V ) be the subvariety of Gr(n, V ) whose points are Lagrangian (i.e. maximal isotropic ) subspaces of V with respect to J. For any point E Lag(n, V ) one define a set U = F Gr(n, V ): V = E F . If F U is also in Lag(n, V ), once ∈ E { ∈ ⊕ } ∈ E the basis e1, ,en of E is fixed, then one can choose a basis f1, ,fn for F1, which is determined by condi- tions J(e {,f )··· = δ }for i, j = 1, ,n. After this choice, (V,J) is{ identified··· with} Q2n with a standard symplectic form i j ij ··· p 0 Idn J0 = . Let G(n) be the symplectic group of level n defined over p-adic field, then the group GQp (n) of Idn 0  −  2n T Q -points of G(n) is given by GQ (n)= Sp(((Q ) ,J ); Q ) := g GL(2n; Q ): J = g J g whose is p p p 0 p { ∈ p 0 0 } denoted by gQp (n).

3.1. Isotropic subspaces and parabolic subgroups. All maximal tori in GQp (n) are conjugate. We fix a Qp-split maximal n × n torus T (Q ) consisting of the diagonal matrices in GQ (n). One can associate any with a Qp ≃ p p t W. For Tn the action of W on Tn is generated by the following transformations[11]: Qp Qp

diag(x1, , xi, xi+1, , xn) 0 σi : ··· ··· 0 diag(x−1, , x−1, x−1 , , x−1)  1 ··· i i+1 ··· n  diag(x , , x , x , , x ) 0 1 ··· i+1 i ··· n , (1 i n 1) 7→ 0 diag(x−1, , x−1 , x−1, , x−1) ≤ ≤ −  1 ··· i+1 i ··· n  diag(x1, , xn−1, xn) 0 σn : ··· 0 diag(x−1, , x−1 , x−1)  1 ··· n−1 n  −1 diag(x1, , xn−1, xn ) 0 ··· −1 −1 . 7→ 0 diag(x1 , , xn−1, xn)  ···  t Therefore W has n elements. The simple roots relative to Tn can be chosen as[11] 2 n! Qp

diag(x1, , xi, xi+1, , xn) 0 −1 αi : ··· ··· xix , (1 i n 1) 0 diag(x−1, , x−1, x−1 , , x−1) 7→ i+1 ≤ ≤ −  1 ··· i i+1 ··· n  diag(x1, , xn) 0 2 αn : ··· x , 0 diag(x−1, , x−1) 7→ n  1 ··· n  each of which determines a unipotent U , and then B Tn U is a where U is generated by all i = Qp U . Any parabolic subgroup of is conjugate to some standard parabolic subgroup which is determined by a subset of i pv GQp (n) α , , α [12]. They can be parameterized as follows. Take an ordered partition (n , ,n ) of k(0 k n), and set { 1 ··· n} 1 ··· l ≤ ≤ g1 ..  .    gl      A B  A B  M =   : g GL(n , Q ), GQ (n k) ,  (g−1)T  i ∈ i p CD ∈ p −   1      .    ..   −1 T    (gl )       CD    t      then P = MB is a standard parabolic subgroup[11]. For the partition k = n1 + + nl(n1 n2 nl), we can consider the filtration (0 W W W W = V ), where W···(i

subspace of V and W (i l)= W ⊥ is the orthogonal complement in V of W withe respect to J, which induces i ≥ 2l−i−1 2l−i−1 a filtration on gQp (n): (0 −2l 2l = gQp (n)) determined by i(Wj ) Wi+j . Then the Lie algebra of corresponding parabolic subgroup⊂ W P is⊂···⊂W exactly . W ⊂ W0

Example 3.1. Let Ak be a k-dimensional isotropic subspace of V (k n), then we have an increasing filtration on V : W = (0 W W W = V ) where W = A , W = A⊥ :=≤ u V : J(u, v)=0 for any v A . This • ⊂ 0 ⊂ 1 ⊂ 2 0 k 1 k { ∈ ∈ k} filtration induces a filtration on gQ (n): = (0 = gQ ) where is determined p W• ⊂ W−2 ⊂ W−1 ⊂ W0 ⊂ W1 ⊂ W2 p Wi by (W ) W . More explicitly, take A = Span e , ,e , A⊥ = Span e , ,e ,f , f , then the Wi j ⊂ i+j k Qp { 1 ··· k} k Qp { 1 ··· n k+1 ··· n} elements in can be represented as the following matrices: Wi 0 0 u 0 00 0 0 −2 =   : u is a symmetric k k  Sym(k; Qp), W 00 0 0 × ≃    00 0 0      0 s u v    0 0 vT 0 −1 =   : u is as that in −2, and s, v are k (n k) matrices , W 00 0 0 W × −    0 0 sT 0    −   ps u v    0 q vT w 0 =   : u,s,v are as those in −1, p is a k k matrix, W 0 0 pT 0 W ×  −  0 r sT qT   − −  q is an (n k) (n k)matrix and w, r are symmetric (n k) (n k) matrices ,  − × − − × − } ps u v t q vT w 1 =   : u,s,v,p,q,w,r are as those in 0, W 0 m pT tT W  − −  mT r sT qT   − −   t is an (n k) k matrix and m is a k (n k) matrix .  − × × − }

We observe that −2 and −1 are both nilpotent subalgebras, and 2/ 1 −2, 1/ 0 −1/ −2. The corre- sponding parabolicW subgroupW whose Lie algebra is consists of all matricesW W that≃ W may beW writtenW ≃ as Wξ = ξWξ ξ ξ where W0 1 2 3 4

Idk 0 0 0 0 A 0 B A B ξ1 =   for GQp (n k), 0 0 Idk 0 CD ∈ −    0 C 0 D     g 0 0 0 0 Idn−k 0 0 ξ2 =   for g GL(k, Qp), 0 0 (g−1)T 0 ∈  00 0 Idn−k      Idk E 0 F T 0 Idn−k F 0 T T ξ3 =   for k (n k) matrices E, F that satisfy EF = F E , 0 0 Idk 0 × −  0 0 ET Id   − n−k  ξ = Id + S with S .  4 n ∈W−2

3.2. Lagrangian subspaces and quadratic forms. Let ℓ1,ℓ2,ℓ3 Lag(n, V ), then one can define a quadratic form Q on ℓ ℓ ℓ as follows: ∈ 1 ⊕ 2 ⊕ 3

Q(z1,z2,z3)= J(z1,z2)+ J(z2,z3)+ J(z3,z1) for z1,2,3 ℓ1,2,3. Assume that Q is non-degenerate. We denote by D(ℓ1,ℓ2,ℓ3) and µ(ℓ1,ℓ2,ℓ3) the and Hasse invariant∈ of this quadratic form Q respectively. From the definition we observe the following obvious properties of µ(ℓ1,ℓ2,ℓ3).

Proposition 3.2. (1) µ(ℓ1,ℓ2,ℓ3) is GQp (n)-invariant. SYMPLECTIC GROUP AND HEISENBERG GROUP IN p-ADICQUANTUMMECHANICS 5

(2) For any permutation of the set 1, 2, 3 , P { } 2 sgn(P)C3n sgn(P)(3n−1) µ(ℓ ,ℓ ,ℓ ) = ( 1, 1)p ( 1,D(ℓ ,ℓ ,ℓ )) µ(ℓ ,ℓ ,ℓ ) P(1) P(2) P(3) − − − 1 2 3 p 1 2 3 µ(ℓ1,ℓ2,ℓ3), p 3 and n is odd; sgn(P) ≥ ( 1,D(ℓ1,ℓ2,ℓ3))p µ(ℓ1,ℓ2,ℓ3), p 3 and n is even; =  2 − sgn(P)C3 ≥  ( 1) n µ(ℓ1,ℓ2,ℓ3), p =2 and n is odd;  − sgn 2 sgn(P) ( 1) (P)C3n ( 1,D(ℓ ,ℓ ,ℓ )) µ(ℓ ,ℓ ,ℓ ), p =2 and n is even. − − 1 2 3 2 1 2 3  × × 2 × × 2 where ( , )p: Q /(Q ) Q /(Q ) 1, 1 stands for the Hilbert symbol. · ·  p p × p p →{ − } More generally for an ordered Lagrangian sequence (ℓ , ,ℓ ) (m 3) on can introduce 1 ··· m ≥ µ(ℓ , ,ℓ ) := µ(ℓ ,ℓ ,ℓ ). 1 ··· m i j k 1≤i

µ(ℓa,ℓb,ℓc,ℓ∗)= µ(ℓ∗,ℓa,ℓb,ℓc)= µ(ℓa,ℓb,ℓ∗,ℓc)= µ(ℓa,ℓ∗,ℓb,ℓc) =(b a,c b) (c b,a c) (a c,b a) − − p − − p − − p (c b,c b) (c a,c a) (b a,b a) − − p − − p − − p =( 1, (a b)(a c)(c b))2 =1. − − − − p 6 SEN HU & ZHIHU

We complete the proof. 

3.3. Latices and maximal compact subgroups. If L is a finitely generated Zp-submodule of V containing a basis of V , it ∗ ∗ is called a lattice in V . The dual lattice L with respect to J of L is defined as L = v V : J(v,u) Zp for u L . If L = L∗, L is called a self-dual lattice. Similarly, one can define the 1-almost self-dual{ ∈ lattice L in (V,J∈ ) if it contains∀ ∈ } a −1 self-dual lattice and J(u, v) p Zp for any u, v L. By choosing a suitable symplectic basis any 1-almost self-dual lattice ∈ ∈ −1 −1 can be reduced to one of Li := Zpe1 Zpen p f1 p fi Zpfi+1 Zpfn(i = 0, ,n), whose ⊕···⊕∗ ⊕ ⊕···⊕ ⊕ ⊕···⊕ ··· corresponding dual latices are given by Li := pe1 pei Zpei+1 Zpen Zpf1 Zpfn. Then we have ∗ ∗ ∗ ⊕···⊕ ⊕ ⊕···⊕ ⊕ ⊕···⊕∗ ∗ i a flag Ln Ln−1 L1 L0 L1 Ln−1 Ln with the property that Li/Li−1 Li−1/Li (Fp) . The stabilizers⊂ of or⊂···⊂∗ in ⊂ are⊂ denoted⊂···⊂ by or⊂ ∗ respectively. They are all maximal≃ paraholic≃ subgroups, Li( Li ) GQp (n) G(Li)( G(Li )) j hence are maximal compact subgroups of GQp (n), and they are not conjugate to each other in GQp (n)[14]. Each maximal ∗ compact subgroup of GQp (n) is conjugate to one of G(Li,n)(= G(Li ,n)). The set of all 1-almost self-dual lattices is given by Λ (n) GQ (n)/G(L ,n). Recursively, one can introduce the l-almost self-dual lattice L which contains an 1 ≃ i p i (l 1)-almost self-dual lattice and satisfies J(u, v) p−lZ for any u, v L. An l-almost self-dual lattice is called the pure F p l-almost− self-dual lattice if it is not an (l 1)-almost∈ self-dual lattice. ∈ − Proposition 3.4. (1) Let L and L′ be pure l-almost and k-almost (k l) self-dual lattices in a 2-dimensional symplectic ′ ≤ l−k space (V,J) respectively, then L is a sub-lattice of L if and only if there exists Θ GL(2, Zp) with detΘ = p up to a constant factor that is a unit such that L =ΘL′. ∈

(2) There is a one-to-one correspondence between the set Λ0(2) GQp (2)/GZp (2) (where GZp (n) = G(n) ≃ 2 4 ∩ GL(n, Zp)) of all self-dual lattices in two dimensional symplectic space (V,J) and Z ⋉ ((Qp) / ), where the 4 ′ ′ ′ ′ ′ ∼ ′ equivalent relation is defined as follows: (Qp) (u,s,r,t) (u ,s , r ,t ) if and only if u u Zp, t t Zp, (r′ r)+ t(u′ u) Z and (s′ s) + (r + r∋′)(u′ u) ∼Z . − ∈ − ∈ − − ∈ p − − ∈ p (3) There are one-to-one correspondences between the sets Λ1(1) and (Z⋉(Qp/Zp)) (Z⋉(Qp/Zp)), and between the 2 4 ⊔ 4 sets Λ1(1) and (Z ⋉((Qp) / )) Λ0(2), where the equivalent relation is defined as follows: (Qp) (u,s,r,t) ′ ′ ′ ′ ′ ∼ ⊔ ′ ′ ′ ′ ′ ∋ ′ ∼ (u ,s , r ,t ) if and only if u u Zp, t t Zp, p(r r)+ t(u u ) p, p(r r)+ t (u u ) Zp and (s′ s) + (r + r′)(u u′) −Z . ∈ − ∈ − − ∈ − − ∈ − − ∈ p a b Proof. (1) L′ is a sub-lattice of L if and only if there exists Θ GL(2, Z ) such that L = ΘL′. For Θ = , we ∈ p c d m   can always make one of a,b,c,d to be a unit via extracting a suitable factor p (m Z+), without lost of generality, we may assume that a or c is a unit. Then we have the following decomposition ∈ a d 1 c 0 1 c 0 1 c − detΘ , c p =1; a b 0 1 1 0 0 c 0 1 | | =       b    c d 1 0 a 0 1 a    , a p =1.  c 1 0 detΘ 0 1 | |  a   a     α 0 thus Θ can be rewritten as the formΘ= A B where A, B SL(2, Z ) and α Z×. Let e,f and e′,f ′ 0 det Θ ∈ p ∈ p { } { }  α  be bases of L and L′ respectively, then e ,f := A−1e, A−1f and e ,f := Be′,Bf ′ are still bases, and e = αe , { 1 1} { } { 2 2} { } 1 2 f = det Θ f . Therefore J(e ,f ) = plµ = detΘJ(e ,f ) = pkν where µ,ν Z× due to the purity of latices, i.e. 1 α 2 1 1 2 2 ∈ p detΘ = pl−kµν−1. ′ (2) Let L0,L0 be two self-dual latices in (V,J), there is a symplectic basis e1,e2,f1,f2 and an element g GQp (2) such that L = Z e Z e Z f Z f and L′ = gL . By ,{ g can} be expressed as ∈ 0 p 1 ⊕ p 2 ⊕ p 1 ⊕ p 2 0 0 pa1 1 u 0 0 1 0 s r pa2 01 0 0 0 1 r t g = h,  p−a1   00 1 0   0 01 0   p−a2   0 0 u 1   0 00 1     −          where a1,a2 Z, u,r,s,t Qp, and h belongs to the group GZp (2) of Zp-points in G(2) which preserves the lattice L0. ′ ∈ ∈ Therefore L0 take the form as L′ =Z pa1 e Z (pa1 ue + pa2 e ) 0 p 1 ⊕ p 1 2 Z (pa1 se + pa1 rue + pa2 re + p−a1 f p−a2 uf ) ⊕ p 1 1 2 1 − 2 Z (pa1 re + pa1 ute + pa2 te + p−a2 f ). ⊕ p 1 1 2 2 ′ ′ −1 ′ ′ On the other hand, if g GQ (2) satisfies g g GZ (2), then g L = gL = L . More precisely, we need ∈ p ∈ p 0 0 0 SYMPLECTIC GROUP AND HEISENBERG GROUP IN p-ADICQUANTUMMECHANICS 7

′ 1 0 s r 1 u 0 0 pa1−a1 ′ − − − 2 0 1 r t 0 1 00 pa2−a  − −     ′  00 1 0 0 0 10 pa1−a1 ′  00 0 1   0 0 u 1   a2−a2       p   1 u′ 0 0  1 0 s′ r′   01 0 0 0 1 r′ t′     × 00 1 0 00 1 0  0 0 u′ 1   00 0 1   −    ′ ′ ′ ′  ′ ′   ′  a1−a1 ′ a2−a2 a2−a2 ′ ′ ′ a1−a1 a −a1 ′ a −a1 a −a2 ( s ru)p + ru p rp + (r + u t )p p 1 u p 1 up 2 − − ′ ′ − ′ − +(s′ + u′r′)pa1−a1 r′upa2−a2 ut′pa2−a2  ′ − ′ −  ′ ′ a2−a2 a1−a1 ′ ′ a −a2 r p (r + ut)p ′ a −a2 a2−a = 0 p 2 − ′ t p 2 tp 2  +u′tpa2−a2 −   ′   a1−a   0 0 p 1 0   ′ ′ ′   0 0 upa1−a1 u′pa2−a2 pa2−a2   −  GZ (2),  ∈ p

′ ′ which implies a1 = a1,a2 = a2 and the equivalent relation. (3) Consider the case n = 1, the 1-almost self-dual lattice L is given by L = Z e p−1Z f , and the corresponding 1 1 p 1 ⊕ p 1 maximal compact subgroup G(L1, 1) is given by

a pb G(L , 1) = : a,b,c,d Z , ad bc =1 1 p−1c d ∈ p −    1 0 α 0 1 p = −1 −1 p 1 0 α × 0 1    α∈Zp   SL(2, Z ), ≃ p z which implies Λ1(1) Λ0(1) Λ0(1), where Λ0(1) Z⋉ (Qp/Zp)[15]. ≃ ≃−1 For the case n =2, we have L1 = Zpe1 Zpe2 p f1 Zpf2, and then F ⊕ ⊕ ⊕

Zp Zp p Zp Zp Zp p Zp G(L1, 2) =  −1 −1 −1  GQp (2) p p Zp p  Zp Zp p Zp  \    1 000  α 1 pb p p a 1 0 0 β 0 1 p Zp × =  −1   −1    : a,b Zp; α, β Zp  .  p Zp 1 a α 00 1 0 ∈ ∈   − −1   Zp Zp 0 1   β   0 0 pb 1        −          Thereby the corresponding Iwasawa decomposition is given by 

pa1 100 0 1 0 00 pa2 u p 0 0 0 1 00 g = h  p−a1   0 0 1 u   s pr 1 0  2 −  p−a   0 0 0 p−1   r t 0 1              for h G(L , 2). L induces a lattice L′ via a symplectic transformation, which takes the form as ∈ 1 1 1 L′ =Z (pa1 e + pa2 ue + p−a1−1(s ur)f + p−a2−1rf ) 1 p 1 2 − 1 2 Z (pa2+1e + p−a1 rf p−a1−1utf + p−a2−1tf ) ⊕ p 2 1 − 1 2 Z p−a1−1f Z ( p−a1−1uf + p−a2−1f ). ⊕ p 1 ⊕ p − 1 2 8 SEN HU & ZHIHU

′ From the same argument in (2), it follows that g L1 = gL1 if and only if ′ pa1−a1 0 0 0 ′ ′ ′ u′pa2−a2−1 upa1−a1−1 pa2−a2 0 0  ′ − ′ ′ ′  a1−a1 ′ a2−a2 a2−a2+1 ′ ′ ′ a1−a1 ′ ′ ′ (ur s)p ru p rp + (r p u t )p a1−a a2−a ′ a1−a ′ ′ ′ p 1 up 2 u p 1  ′ − ′ ′ a1−a−1 ′ a2−a2 − ′ a2−−a2   +(s u r )p + r up +ut p −  − ′ ′  ′ a2−a2 a1−a1−1 ′ ′ ′   r p + (ut rp)p ′ a2−a a −a2 a2−a   ′− t p 2 tp 2 0 p 2   u′tpa2−a2−1 −   −  Sp ((Q4,J ); Z ),  ∈ p 1 p 0 0 p−1 0 0 0 01 where J = . 1  p−1 0 0 0  −  0 1 0 0   −  Similarly, for L = Z e Z e p−1f p−1f we have 2 p 1 ⊕ p 2 ⊕ 1 ⊕ 2 Zp Zp p p Zp Zp p p G(L2, 2) =  −1 −1  GQp (2) p p Zp Zp −1 −1  p p Zp Zp  \   1 000 α  1 b p p a 1 0 0 β 0 1 p p × =       : a,b Zp; α, β Z  p−1 p−1 1 a α−1 00 1 0 ∈ ∈ p  −   p−1 p−1 0 1   β−1   0 0 b 1        −   1 000  α   0 10 0 1 b p p    a 1 0 0 β 1 00 0 0 1 p p   p−1 p−1 1 a   α−1   0 00 1   00 1 0   − [  p−1 p−1 0 1   β−1   0 01 0   0 0 b 1         −  : a,b Z ; α, β Z× .         ∈ p ∈ p From these explicit expressions we can deduce the conclusions. 

4. p-ADIC HEISENBERG GROUP The standard quantum mechanics starts with a representation of the well-known Heisenberg commutation relation [ˆq, pˆ] iId where q,ˆ pˆ are unbounded self-adjoint linear operator on a Hilbert space and the domain of [ˆq, pˆ] is a dense subset⊂ in H ∞ and [ˆq, pˆ] = iId. Let E , <λ< be the spectrum of pˆ, i.e. p = λdE . Define a family of unitary H |Dom([ˆq,pˆ]) { λ −∞ ∞} −∞ λ operators U(t) = eiptˆ := ∞ eitλdE ,t R. Similarly one may define V (t) = eiqtˆ . Then the Heisenberg commutation −∞ λ R relation is equivalent to the relation ∈ R U(t)V (s)= eitsV (s)U(t). (4.1) eq:a i eq:a Another approach is to define a unitary operator Q(z)= U(t)V (s)e− 2 ts where z = (t,s) R2. Then (4.1) is equivalent to the relation ∈ i ′ Q(z)Q(z′)= e 2 J0(z,z )Q(z + z′), (4.2) eq:b eq:a eq:b ′ ′ ′ where J0(z,z )= ts t s.(4.1)or(4.2) is called Weyl formulation of quantum mechanics where one represents Heisenberg group rather than its algebra.− It provides an appropriate framework for a generalization to p-adic variables.

Definition 4.1. The p-adic Heisenberg group Hp(V ) over a 2n-dimensional (V,J) is the of V by a unit circle S1 in the field C, i.e. the set of pairs (u; α): u V, α S1 with the composition law { ∈ ∈ } 1 (u; α) (v; β) = (u + v; αβχ( J(u, v))). (4.3) c · 2 4.1. Schrödinger representations and Weyl operators. Under the Schrödinger picture, the Heisenberg group H(V ) can be realized on the group of unitary operators acting on the space (Qn) as follows. Let us choose a symplectic basis such U D p that (V,J) (Q2n,J ), and equip with a weak operator topology, then we define a continuous homomorphism called the ≃ p 0 U Schrödinger representation Φ: H(V ) via Φ(g)[ψ](ξ)= αT [ψ](ξ), where →U z 1 1 T [ψ](ξ)= ψ(ξ + x)χ( (y ξ + x y )) = ψ(ξ + x) χ(y ξ + x y ) (4.4) 1 z i i 2 i i i i 2 i i i i X Y SYMPLECTIC GROUP AND HEISENBERG GROUP IN p-ADICQUANTUMMECHANICS 9

for g = (z = (x , , x ,y , y ); α) H(V ) in terms of the chosen symplectic basis of V and ξ = (ξ , , ξ ) Qn. 1 ··· n 1 ··· n ∈ 1 ··· n ∈ p Proposition 4.2. The Schrödinger representation is an irreducible and admissible representation.

n Proof. Suppose W is a non-tivial H(V )-invariant subspace of (Qp ). One can choose an orthonormal basis ψα for W W D { } such that each element belongs to can be written as the linear combination of finite ψαs. Let Eα be the support1 of ψα, n n E = α Eα and (E) be the space of functions vanishing on Qp E in (Qp ). It follows from the definition (4.4) that D \ D n fα(ξ)= T(0,y)[ψ](ξ)= χ( i yiξi)ψα(ξ) W for any y = (y1, ,yn) Qp . Obviously, W (E), moreover we claim S ∈ ··· ∈ ⊂ D n that W = (E). Otherwise there is a non-zero function g (E) such that (fα,g)= Qn fα(ξ)g(ξ)d ξ = F [ψαg](y)=0 D P ∈ D p n n for α, which implies that ψα(ξ)g(ξ) vanishes on Qp . Hence g has to be zero on ER , but g 0 on Qp E, so g = 0, ∀ n ≡ \ which exhibits a contradiction. So far, we only need to show E = Qp up to a zero measure set. Indeed, if ψ(ξ) (E), n ∈ D then ψ(ξ + x) (E) for any x = (x1, , xn) Qp , namely ψ(ξ + x) (E) vanishes outside E. Therefore E is ∈ D ··· ∈ n ∈ D quasi-invariant with respect to the translation. It is known that Qp as the Harr measure space is ergodic with respect to the translation group, then Qn E is a zero measure set since E cannot be a zero measure set. p \ Therefore the Schrödinger representationp is an irreducible smooth representation, hence to show it is admissible we only need to show it is supercuspidal[16]. Indeed, since Schrödinger representation (Φ, (Qn)) and the induced representation D p (Φ∗, (( (Qn))∗) are both smooth, if one fixes ψ (Qn), Ψ ( (Qn))∗, then there exist a compact open subgroup K of D p ∈ D p ∈ D p H(V ) such that ψ ( (Qn))K , Ψ (( (Qn))∗)K and Ψ(Φ(k gk )ψ) = Ψ(Φ(g)ψ) for any k , k K, and g H(V ), ∈ D p ∈ D p 1 2 1 2 ∈ ∈ where ( (Qn))K = ψ (Qn) : Φ(g)ψ = ψ for any g K . Take g = ((x, y); 1) and k ( or k ) = ((x′,y′); 1) D p { ∈ D p ∈ } 1 2 with non-zero x′,y′, then Ψ(Φ(((x, y); 1))ψ) = χ( 1 (x′ y x y′))Ψ(Φ(((x + x′,y + y′); 1))ψ) = χ(x′ y + 1 x′ 2 · − · · 2 · y′)Ψ(Φ(((x, y); 1))ψ)= χ(y′ x + 1 x′ y′)Ψ(Φ(((x, y); 1))ψ) which shows the supercuspidality.  · 2 · mv Theorem 4.3. (Non-Archimedean Stone-von Neumann Theorem[17]) Any smooth representation Φ of the Heisenberg group H(V ) satisfying Φ(0, α)= αId decomposes into the direct sum of irreducible representations equivalent to the Schrödinger representations.

n ∗ n n Let A : (Qp ) (Qp ) be a linear operator, then we define a bilinear functional LA on (Qp ) associated to A as follows: D → D D

1 2 2 2 ∗ LA(ψ, ϕ)= α χ 1 ( (x y )) Φ (g) A(ψ), ϕ dµg 8 i − i h ◦ i i Z X for ψ, ϕ (Qn), where the linear operator Φ(g)∗ : ∗(Qn) ∗(Qn) is induced by Φ(g), , stands for the pairing ∈ D p D p → D p h i between ∗(Qn) and (Qn), and dµ denotes the invariant measure on H(V ). Hence there exists a a linear operator D p D p g ak P : (Qn) ∗(Qn) such that L (ψ, ϕ)= P ψ, ϕ [10]. A D p → D p A h A i

Proposition 4.4. LPA = CLA, where C is a constant. Proof. According to the definition, we calculate

1 ′ 2 ′ 2 ′ 2 ′ ∗ ∗ L ∗ (ψ, ϕ)= (α ) χ 1 ( ((x ) (y ) )) Φ(g ) Φ(g) PA(ψ), ϕ dµg′ Φ(g) ◦PA 8 i − i h ◦ ◦ i i Z X 1 ′ 2 ′ 2 ′ 2 ′ = (α ) χ 1 ( ((x ) (y ) )) PA(ψ), Φ(g) Φ(g )(ϕ) dµg′ 8 i − i h ◦ i i Z X 1 1 ′ 2 ′′ 2 ′ 2 ′ 2 ′′ 2 ′′ 2 ′ ∗ ∗ ′′ ∗ = (α ) (α ) χ 1 ( ((x ) (y ) + (x ) (y ) )) Φ(g ) Φ(g) Φ(g ) A(ψ), ϕ dµg′ dµg′′ . 8 i − i i − i h ◦ ◦ ◦ i i Z X ′ ′′ ′ ˜ ′′ ′ ˜ ′′ Let g˜0 = gg , g˜ = g g˜0; and Xi = xi + xi, Xi = xi + Xi, Yi = yi + yi, Yi = yi + Yi, then we arrive at

∗ LΦ(g) ◦PA (ψ, ϕ)

1 1 ′′ 2 ′ 2 1 ˜ 2 ˜ 2 2 2 ∗ = (α ) (α ) χ ( ((Xi Xi) (Yi Yi) + (Xi xi) (Yi yi) )) Φ(˜g) A(ψ), ϕ dµg˜dµg˜0 8 − − − − − − h ◦ i i Z X 1 − 2 1 2 2 1 2 ˜ ˜ 2 ˜ ˜ =α χ ( (x y )) χ ( (X (Xi + xi Yi yi)Xi Y + (Yi + yi Xi xi)Yi))dµg˜0 8 i − i 4 i − − − − i − − i i X Z Z X 1 1 ′′ ′ ′ ′′ ′′ ′ ′ ′′ ′′ ′ 2 2 2 ∗ (αα α χ( (xiy + x yi + x y yix y xi y x ))) χ 1 ( (X˜ Y˜ )) Φ(˜g) A(ψ), ϕ dµg˜ · 2 i i i i − i − i − i i 8 i − i h ◦ i i i X X =CΘ(g)LA(ψ, ϕ), 10 SEN HU & ZHIHU

where − 1 2 2 Θ(g)= α 2 χ 1 ( (x y )), 8 i − i i X1 1 1 λp( 4 )λp( 4 ) 1 1 1 C = Vol(S ) − = Vol(S ) 4 pλp( )λp( ). 1 | | 4 −4 |− 16 |p ∗ q  Consequently, we find that LΦ(g) ◦PA = CΘ(g)LA, which implies that LPA = CLA when one takes g = Id. We can define the so-called symplectic Fourier transformation for ψ (Q2n) as follows: ∈ D p ′ ′ 2n ′ Fs[ψ](z)= ψˇ(z)= χ(J0(z,z ))ψ(z )d z , Z which is related with the usual Fourier transformation via the formula Fs[ψ](J0z)= F [ψ](z). Therefore Fs as an operator on 2n 2 2n 2n (Qp ) extends into a unitary operator on L (Qp ). The Weyl operator Wf associated to the symbol f (Qp ) is defined Dby ∈ D

ˇ 2n Wf [ψ](ξ)= f(z)T(z,1)[ψ](ξ)d z Z 1 = fˇ(z)ψ(ξ + x)χ( (y ξ + x y ))d2nz. i i 2 i i i Z X Proposition 4.5. W is a linear and continuous operator on (Qn), namely W [ψ] (Qn) for ψ (Qn). f D p f ∈ D p ∈ D p ˇ 2n ′ 2n l Proof. From f (Qp ), it follows that there exists z = (0, , 0,a1, ,an) Qp with ai p = p (i = 1, ,n) for ∈ D ˇ ˇ ′ ··· ··· ∈ | | −l ··· an integer l such that f(z)= f(z + z ). Then since Wf [ψ](ξ) = χ( i ξiai)Wf [ψ](ξ), and since if ξ p >p , there exists | |n k 1, ,n such that ξkak p > 1, thus χ( i ξiai) =1, we observe that supp(Wf [ψ]) (B(0; l)) . To show Wf [ψ] is a locally∈{ ··· constant} function,| we consider| 6 P ⊂ P N ˇ 1 2n Wf [ψ](ξ) := f(z)ψ(ξ + x)χ( (yiξi + xiyi))d z 2n 2 (B(0;−N)) i Z X for an integer N. Let l Z be the largest characteristic number associated with ψ. Then for any ξ′ (B(0; N))n we have W N [ψ](ξ + ξ′) = W N∈[ψ](ξ) if N > l because y ξ′ max y ξ′ 1 and ψ(ξ + x +∈ξ′) = ψ(ξ + x) when f f − | i i i|p ≤ {| i i|p} ≤ N > l. Consequently, W [ψ] = lim W N [ψ] is locally constant.  − f N→∞ f P n Proposition 4.6. (1) Wf [ψ](ξ)= KWf (ξ, η)ψ(η)d η, where 1 K R(ξ, η)= f( (η + ξ),y)χ( (η ξ )y )dny Wf 2 i − i i i Z X is called the kernel function of Weyl operator. (2) Wf [ψ] L2 F [f] L1 ψ L2 . (3)|| For f,g || ≤(Q ||n), we|| have|| the|| composition law K = K , where ∈ D p Wf ◦Wg Wh h(z)= c f(z + z′)g(z + z′′)χ( 2J (z′,z′′))d2nz′d2nz′′ (Qn) − 0 ∈ D p Z 1, p 3; with c = 1 ≥ n , p =2.  4 Proof. (1) Let η = ξ + x, then we have 1 W [ψ](ξ)= fˇ(η ξ,y)ψ(η)χ( y (η + ξ ))dnηdny f − 2 i i i i Z X 1 = f(z′)ψ(η)χ( (η ξ )y′)χ( y (η + ξ 2x′ ))d2nz′dnηdny i − i i 2 i i i − i i i Z X X 1 = f(z′)ψ(η)χ( (η ξ )y′)δ( (η + ξ 2x′ ))d2nz′dnη i − i i 2 i i − i i Z X 1 = f( (η + ξ),y′)ψ(η)χ( (η ξ )y′)dny′dnη 2 i − i i i Z X SYMPLECTIC GROUP AND HEISENBERG GROUP IN p-ADICQUANTUMMECHANICS 11

ak where we have applied the Fourier transform of Dirac distribution δ for the third equality[10]: F [δ]=1ak and F [1] = δ. 1 2n 2n 1 2n (2) First we note that F [f] L (Qp ) since F [f] belongs to (Qp ) which is dense in L (Qp )[10]. By manipulation of the Fourier transform of Dirac∈ distribution once again, we expressD the kernel function as 1 K (ξ, η)= f(k,y)χ( (η ξ )y )χ( x (2k η ξ ))dnkdnydnx Wf i − i i 2 i i − i − i i i Z X X 1 = F [f](x, η ξ)χ( x (η + ξ ))dnx, − − 2 i i i i Z X which leads to the desired inequalities

W [ψ](ξ) 2dnξ ( K (ξ, η) dnη)( K (ξ, η) ψ(η) 2dnη)dnξ | f | ≤ | Wf | | Wf | · | | Z Z Z Z ( F [f](ξ, η) dnξdnη)2 ψ(η) 2dnη. ≤ | | | | Z Z n (3) One can easily check that KWf ◦Wg (ξ, η)= KWf (ξ, ζ)KWg (ζ, η)d ζ. Then since

n R KWf (ξ, ζ)KWg (ζ, η)d ζ Z 1 1 = f( (ξ + ζ), x)g( (ζ + η),y)χ( (ζ ξ )x )χ( (η ζ )y )dnxdnydnζ, 2 2 i − i i i − i i i i Z X X we get 1 1 h(ξ, η)= K (ξ x, ξ + x)χ( η x )dnx Wf ◦Wg − 2 2 − i i i Z X 1 1 1 1 = f( (ξ x + ζ), α)g( (ξ + x + ζ),β) 2 − 2 2 2 Z 1 1 χ( (ζ ξ + x)α )χ( (ξ ζ + x)α )χ( η x )dnαdnβdnζdnx i − i 2 i i − i 2 i − i i i i i X X X 1 = −n f(ξ + u, η + α′)g(ξ + v, η + β′)χ(2 (v α′ u β′))dnα′dnβ′dnudnv, |− 4|p i i − i i i Z X where the constant 1 −n = 4 n in front of the integral comes from the variable changing u = 1 ( 1 x ξ + η), | − 4 |p | |p 2 − 2 − v = 1 ( 1 x ξ + η), α′ = α η and β′ = β η.  2 2 − − − 4.2. Induced representations and Maslov indices. Let Γ be an abelian subgroup of the Heisenberg group H(V ), then we have a smooth representation Υ(Γ, ) of H(V ) induced by the given unitary character of Γ restricting to the identity on S1 C C H(V ) the center of H(V ) and being locally constant restricted on V , namely Υ(L, ) = IndΓ , which can be realized as follows. Let H(Γ, ) be the set of complex valued functions on H(V ) which are locallyC constantC compactly supported when restricted on V andC satisfy the conditions ψ(γg) = (γ)ψ(g) for any g H(V ),γ Γ. Then the induced representation C H ∈ ∈ Υ(Γ, ) is defined to be the representation of H(V ) in (Γ, ) given by right translations: Υ(Γ, )(g0)[ψ](g)= ψ(gg0)hofor C C 1 C ψ H, g0,g H(V ). Obviously, Υ(Γ, )(α) = αId for α S . One can show that (Υ(Γ, ), H) is an irreducible[18] and∈ admissible∈ representation. Stone-von NeumannC theorem implies∈ that every induced representationC Υ(Γ, ) is isomorphic to the Schrödinger representation, thus there exists a unique unitary isomorphism Θ(Γ, ;Γ′, ′): H(Γ, C) H(Γ′, ′) determined by the relation Θ(Γ, ;Γ′, ′)Υ(Γ, )Θ−1(Γ, ;Γ′, ′) = Υ(Γ′, ′) up to aC unit scalarC relatedC to→Γ, Γ′. ThisC isomorphism is called the intertwiningC operatorC betweenC theC twoC correspondingC representations. For example, we can take Γ = (L, S1) =: Γ(L) or Γ = (ℓ, S1) =: Γ(ℓ) for any self-dual lattice L or Lagrangian subspace ℓ in (V,J). By the restriction from H(V ) to V , the function ψ H(Γ, ) descends to a function on V denoted still by ψ that 1 ∈ C 1 satisfies ψ(z + γ) = χ( 2 J(z,γ)) ((γ; 1))ψ(z) for z V,γ L or ℓ, and Υ(Γ, )(g0)[ψ](z) = αχ( 2 J(z, w))ψ(z + w) C ∈ ∈ C ′ for g0 = (w; α) H(V ). For a self-dual lattice L, there exist two transversal Lagrangian subspaces ℓ and ℓ , where the ∈ ′ ′ transversality condition means that ℓ ℓ = 0 , such that L = ℓ L ℓ L. The characters ℓ, L of Γ(ℓ) and Γ(L) are specified such that ((ℓ L;1)) = ∩ ((ℓ {L;} 1)). ∩ ⊕ ∩ C C CL ∩ Cℓ ∩ Proposition 4.7. The isomorphism Θ(Γ(ℓ), ; Γ(L), ): H(Γ(ℓ), ) H(Γ(L), ) is given by Cℓ CL Cℓ → CL Θ(Γ(ℓ), ℓ; Γ(L), L)[ψ](g) L(( u; 1))ψ((u; 1) g). C C ∼ ′ C − · u∈Xℓ ∩L where means the equality up to a constant of modulus one related to L and ℓ. ∼ 12 SEN HU & ZHIHU

Proof. We first need to show Θ(Γ(ℓ), ; Γ(L), )[ψ] H(Γ(L), ) for ψ H(Γ(ℓ), ). If one chooses a suitable Cℓ CL ∈ CL ∈ Cℓ symplectic basis e1, ,en,f1, ,fn of V such that L = Zpe1 Zpen Zpf1 Zpfn, ℓ = Qpe1 Qpen and ℓ′ = Q f { ···Q f , then··· for g }= ((x, y); α) we express ⊕···⊕ ⊕ ⊕···⊕ ⊕···⊕ p 1 ⊕···⊕ p n 1 Θ(Γ(ℓ), ; Γ(L), )[ψ](g) α (x)χ( x y ) ( u)χ( u x )ψ(u + y), Cℓ CL ∼ Cℓ − 2 i i CL − − i i i Zn i X uX∈ p X where ℓ(x) := ℓ(((x, 0); 1)), L(u) := L(((0,u); 1)) and ψ(u + y) := ψ(((0,u + y); 1)). The condition that Θ(Γ(ℓ)C, ; Γ(L), C )[ψ](γg) = C (γ)Θ(Γ(ℓC), ; Γ(L), )[ψ](g) for any γ Γ(L) can be easily verified. Obviously Cℓ CL CL Cℓ CL ∈ Θ(Γ(ℓ), ℓ; Γ(L), L)[ψ] is locally constant restricted on V since the characteristic functions associated with locally constant C C n n functions χ( i uixi) and ψ(u + y) on Qp can be taken independent of u. On the other hand, there exists a Zp with l − ∈ a p = p for some integer l such that Θ(Γ(ℓ), ℓ; Γ(L), L)[ψ](g) = L((0,a); 1)χ( i xiai)Θ(Γ(ℓ), ℓ; Γ(L), L)[ψ](g), | | P n C C C C C meanwhile, we have y Qp : Θ(Γ(ℓ), ℓ; Γ(L), L)[ψ]((x, y); α) = 0 Zn y u : ψ(((0,y); 1)) = 0 , { ∈ C C 6 } ⊂ Pu∈ p { − 6 } the latter set is bounded in Qn. Therefor Θ(Γ(ℓ), ; Γ(L), )[ψ] is compactly supported on V . Next we should prove p Cℓ CL S Θ(Γ(ℓ), ℓ; Γ(L), L) is surjective. To show it, we define the intertwining operator Θ(Γ(L), L; Γ(ℓ), ℓ): H(Γ(L), L) H(Γ(ℓ), C ) by C C C C → Cℓ Θ(Γ(L), ; Γ(ℓ), )[ψ](g) (( u; 1))ψ((u; 1) g)dµ CL Cℓ ∼ Cℓ − · ℓ/ℓ∩L Zℓ/ℓ∩L 1 n = α ℓ( u)χ( uiyi)ψ(u + x, y)d u n n C − 2 Qp /Zp i Z X 1 1 n = α ℓ(x)χ( xiyi) ℓ( u)χ( uiyi)ψ(u,y)d u, C − 2 Qn Zn C − 2 i p / p i X Z X where regarding u as a representative of an equivalent class in ℓ/ℓ L makes sense, and ψ(u + x, y) := ψ(((u + x, y); 1)). ∩ The similar arguments implies that Θ(Γ(L), L; Γ(ℓ), ℓ)[ψ] H(Γ(ℓ), ℓ). We also have to show Θ(Γ(ℓ), ℓ; Γ(L), L) Θ(Γ(L), ; Γ(ℓ), ) Id. Indeed, for ψ CH(Γ(L),C ) we∈ calculate C C C ◦ CL Cℓ ∼ ∈ CL Θ(Γ(ℓ), ; Γ(L), ) Θ(Γ(L), ; Γ(ℓ), )[ψ](g) Cℓ CL ◦ CL Cℓ 1 n α ℓ( u)χ( (uivi + uiyi))ψ(u + x, y)d u ∼ Qn Zn C − 2 Zn p / p i vX∈ p Z X = ψ((x, y); α). Finally, the isometry property of these can been immediately seen from the Parseval-Steklov equality for Fourier transformations. 

Corollary 4.8. Let ℓ ,ℓ be two Lagrangian subspaces in (V,J), and the characters , of Γ(ℓ ), Γ(ℓ ) be chosen to 1 2 Cℓ1 Cℓ1 1 2 satisfies ℓ1 ((ℓ1 ℓ2;1)) = ℓ2 ((ℓ1 ℓ2; 1)), then the isomorphism Θ(Γ(ℓ1), ℓ1 ; Γ(ℓ2), ℓ2 ): H(Γ(ℓ1), ℓ1 ) H(Γ(ℓ2), ℓ2 ) is given byC ∩ C ∩ C C C → C

Θ(Γ(ℓ ), ; Γ(ℓ ), )[ψ](g) (( u; 1))ψ((u; 1) g)dµ (4.5) eq:p 1 Cℓ1 2 Cℓ2 ∼ Cℓ2 − · ℓ2/ℓ1∩ℓ2 Zℓ2/ℓ1∩ℓ2 n m 1 m = α ℓ2 (x)χ( xiyi) ℓ2 ( u)χ( uixi)ψ(u + y)d u, C − 2 Qm C − − i=1 p i=1 X Z X where the measure µ on ℓ /ℓ ℓ has to be suitably chosen to guarantee the unitarity of intertwining operators, ℓ2/ℓ1∩ℓ2 2 1 ∩ 2 and the second equality is expressed in terms of the suitable symplectic basis of (V,J) with m = n dimQp ℓ1 ℓ2 and (u)= (((u , ,u , 0, , 0); 1)), ψ(u + y)= ψ(((u , ,u , 0, , 0,y , ,y ); 1)). − ∩ Cℓ2 Cℓ2 1 ··· m ··· 1 ··· m ··· 1 ··· n Proof. There exists a symplectic basis e , ,e ,f , ,f such that ℓ = Q e Q e and ℓ = Q f { 1 ··· n 1 ··· n} 1 p 1 ⊕···⊕ p n 2 p 1 ⊕ Qpfm Qpem+1 Qpen. We fix a self-dual lattice L = Zpe1 Zpen Zpf1 Zpfn. Then ℓ1 ℓ2 = Q e ⊕ Q e···⊕and L = Z e Z e Z f ⊕···⊕Z f are Lagrangian⊕ subspace⊕···⊕ and self-dual lattice∩ in p m+1 ···⊕ p n 0 p m+1 ···⊕ p n ⊕ p m+1 ···⊕ p n (V0 = Qpem+1 Qpen Qpfm+1 Qpfn,J V0 ) respectively. The isomorphism Θ(Γ(ℓ1), ℓ1 ; Γ(ℓ2), ℓ2 ) can be constructed via the···⊕ composition⊕ ···⊕ | C C

Θ(Γ(ℓ ), ; Γ(ℓ ), ) Θ(Γ(L), ; Γ(ℓ ), ) Θ(Γ(ℓ ), ; Γ(L), ), 1 Cℓ1 2 Cℓ2 ∼ CL 2 Cℓ2 ◦ 1 Cℓ1 CL SYMPLECTIC GROUP AND HEISENBERG GROUP IN p-ADICQUANTUMMECHANICS 13

where the character L is fixed with the properties that L((ℓ1,2 L;1)) = ℓ1,2 ((ℓ1,2 L; 1)). On the other hand, we observe that C C ∩ C ∩ Θ(Γ(ℓ ), ; Γ(L), ) 1 Cℓ1 CL ′ ′ Θ(Γ(ℓ ℓ ), ; Γ(L ), ) Θ(Γ(ℓ ), ′ ; Γ(L ), ′ ), ∼ 1 ∩ 2 Cℓ1 |ℓ1∩ℓ2 0 CL|L0 ⊗ 1 Cℓ1 |ℓ1 CL|L Θ(Γ(L), ; Γ(ℓ ), ) CL 2 Cℓ2 ′ ′ Θ(Γ(L ), ; Γ(ℓ ℓ ), ) Θ(Γ(L ), ′ ; Γ(ℓ ), ′ ). ∼ 0 CL|L0 1 ∩ 2 Cℓ2 |ℓ1∩ℓ2 ⊗ CL|L 2 Cℓ2 |ℓ2 ′ ′ ′ where ℓ1 = ℓ1/ℓ1 ℓ2 = Qpe1 Qpem, ℓ2 = ℓ2/ℓ1 ℓ2 = Qpf1 Qpfm and L = L/L0 = Zpe1 Z e Z f ∩ Z f . As⊕···⊕ a consequence, we arrive at∩ ⊕···⊕ ⊕···⊕ p m ⊕ p 1 ⊕···⊕ p m Θ(Γ(ℓ ), ; Γ(ℓ ), )[ψ](g) 1 Cℓ1 2 Cℓ2 ′ ′ ′ ′ Θ(Γ(L ), ′ ; Γ(ℓ ), ′ ) Θ(Γ(ℓ ), ′ ; Γ(L ), ′ )[ψ](g) ∼ CL|L 2 Cℓ2 |ℓ2 ◦ 1 Cℓ1 |ℓ1 CL|L

′ ′ ′ ′ ′ ℓ2 ℓ2 (( u; 1)) L L (( v; 1))ψ((v; 1) (u; 1) g)dµℓ2/ℓ2∩L ′ ′ ′ ∼ ℓ2/ℓ2∩L C | − ′ ′ ′ C | − · · Z v∈LX/L ∩ℓ1

′ ′ = ℓ2 ℓ2 (( u; 1))ψ((u; 1) g)dµℓ2 . ′ C | − · Zℓ2 Thus, we complete the proof.  eq:p Definition 4.9. The isomorphism as the r.h.s. of (4.5) is called the canonical isomorphism between the representation spaces

H(Γ(ℓ1), ℓ1 ) and H(Γ(ℓ2), ℓ2 ), and denoted by Θc(Γ(ℓ1), ℓ1 ; Γ(ℓ2), ℓ2 ). For a triple (ℓ1,ℓ2,ℓ3) of Lagrangian subspaces C C0 C C in (V,J) and the characters ((ℓi;1)) = 1(i =1, 2, 3), we have Cℓi Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 ) Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 ) Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 ) c 3 Cℓ3 1 Cℓ1 ◦ c 2 Cℓ2 3 Cℓ3 ◦ c 1 Cℓ1 2 Cℓ2 =α(ℓ1,ℓ2,ℓ3)Id, (4.6)

1 and the coefficient α(ℓ1,ℓ2,ℓ3) S is called the Maslov index associated with (ℓ1,ℓ2,ℓ3), or equivalently, the Maslov index is determined by ∈ Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 ) Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 )= α(ℓ ,ℓ ,ℓ )Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 ). (4.7) c 2 Cℓ2 3 Cℓ3 ◦ c 1 Cℓ1 2 Cℓ2 1 2 3 c 1 Cℓ1 3 Cℓ3 Proposition 4.10. Let ℓ1,ℓ2,ℓ3,ℓ4 be four Lagrangian subspaces in (V,J), then the Maslov index is given by 1 α(ℓ ,ℓ ,ℓ )= χ( J(w, v))dµ dµ , 1 2 3 2 ℓ3 /ℓ2∩ℓ3 ℓ2/ℓ2∩ℓ1 Zv∈ℓ3/ℓ2∩ℓ3,w∈ℓ2/ℓ1∩ℓ2,v+w∈ℓ1 and it has the following properties:

(permutation relations) α(ℓ1,ℓ2,ℓ3)= α(ℓ1,ℓ3,ℓ2), α(ℓ1,ℓ2,ℓ3)= α(ℓ2,ℓ3,ℓ1), • (cocycle relations) α(ℓ ,ℓ ,ℓ )α(ℓ ,ℓ ,ℓ )α(ℓ ,ℓ ,ℓ )α(ℓ ,ℓ ,ℓ )=1. • 1 2 3 1 3 4 2 4 3 2 1 4 Proof. According to the definitions, for any ψ H(Γ(ℓ ), 0 ) and g = (z; α) H(V ), we have ∈ 1 Cℓ1 ∈ Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 ) Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 ) Θ (Γ(ℓ ), 0 ; Γ(ℓ ), 0 )[ψ](g) c 3 Cℓ3 1 Cℓ1 ◦ c 2 Cℓ2 3 Cℓ3 ◦ c 1 Cℓ1 2 Cℓ2

= dµℓ1/ℓ1∩ℓ3 dµℓ3/ℓ2∩ℓ3 dµℓ2/ℓ2∩ℓ1 Zℓ1/ℓ1∩ℓ3 Zℓ3/ℓ2∩ℓ3 Zℓ2/ℓ1∩ℓ2 1 1 1 αχ( J(u,z)+ J(v,u + z)+ J(w,u + v + z))ψ(u + v + w + z) 2 2 2

= dµℓ1/ℓ1∩ℓ3 dµℓ3/ℓ2∩ℓ3 dµℓ2/ℓ2∩ℓ1 Zℓ1/ℓ1∩ℓ3 Zℓ3/ℓ2∩ℓ3 Zℓ2/ℓ1∩ℓ2 1 1 αχ( J(w, v)+ J(v + w,u)+ J(v + w,z))ψ(v + w + z) 2 2 1 =( χ( J(w, v))dµ dµ )ψ(g), 2 ℓ3 /ℓ2∩ℓ3 ℓ2/ℓ2∩ℓ1 Zv∈ℓ3/ℓ2∩ℓ3,w∈ℓ2/ℓ1∩ℓ2,v+w∈ℓ1

where ψ has been viewed as a functionz on V in the first and the second equalities. The properties of Maslov index can be easily deduced from the definition[15].  14 SEN HU & ZHIHU

Example 4.11. Let us consider an example where ℓ1,ℓ2,ℓ3 Lag(1, V ). There are two non-trivial cases. (i) If (ℓ ,ℓ ,ℓ ) = (ℓ ,ℓ ,ℓ ),a = b = c, we can directly∈ calculate 1 2 3 a b c 6 6 1 α(ℓ ,ℓ ,ℓ )= χ( J(w, v))dµ dµ a b c 2 ℓc ℓb Zv∈ℓc,w∈ℓb,v+w∈ℓa (c b)(a c) (c b)(a c) = χ( − − x2)dx/ χ( − − x2)dx 2(b a) | 2(b a) | ZQp − ZQp − (c−b)(a−c) λp( 2(b−a) ), p> 3; = 1 (c−b)(a−c) ( 2 λ2( 2(b−a) ), p =2. (ii) If (ℓ ,ℓ ,ℓ ) = (ℓ ,ℓ ,ℓ ),a = b, we similarly have 1 2 3 a b ∗ 6 a b a b α(ℓ ,ℓ ,ℓ )= χ( − x2)dx/ χ( − x2)dx a b ∗ 2 | 2 | ZQp ZQp λ ( a−b ), p> 3; = p 2 1 λ ( a−b ), p =2.  2 2 2 REFERENCES [1] I. V. Volovich, Number theory as the ultimate physical theory, Preprint No. TH 4781/87, CERN, 1987. [2] B. Dragovich, A. Y. Khrennikov, S. V. Kozyrev, and I. V. Volovich, On p-adic mathematical physics, p-Adic Numbers, Ultrametric Analysis, and Applications 1 (1), 1-17 (2009). [3] L. Brekke, P. G. O. Freund, M. Olson, and E. Witten, Non-Archimedean string dynamics, Nucl. Phys. B 302 (3), 365-402 (1988). [4] M. Frau, L. Gallot, A. Lerda, and P. Strigazzi, Stable non-BPS D-branes in type I string theory, Nucl. Phys. B, 564 (1), 60-85 (2000). [5] P. G. Freund and M. Olson, Non-archimedean strings, Phys. Lett. B 199(2), 186-190 (1987). [6] V. S. Vladimirov and I. V. Volovich, p-Adic quantum mechanics, Commun. Math. Phys. 123, 659-676 (1989) [7] A. Y. Khrennikov, p-Adic quantum mechanics with p-adic valued functions, Journal of Math. Phys. 32(4), 932-937 (1991). [8] N. Koblitz, p-Adic numbers, p-adic analysis, and zeta-functions (Second Edition), Graduate Texts in Mathematics 58, Springer-Verlag, 1984. [9] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic analysis and mathematical physics, World Scientific Publishing, 1994. [10] S. Albeverio, A. Y. Khrennikov, and V. M. Shelkovich, Theory of p-adic distributions: linear and nonlinear models, Cambridge University Press, 2010. [11] M. Tadic,´ Representaion of p-adic symplectic groups, Compositio Math. 90, 123-181 (1994). [12] V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Academic Press, 1994. [13] J. P. Serre, A course in arithmetic, Graduate Texts in Mathematics 7, Springer-Verlag, 1973. [14] L. Ji, Buildings and their applications in geometry and topology, Asian J. Math. 10 (2006), 11-80. [15] E. I. Zelenov, p-Adic Heisenberg group and Maslov index, Commun. Math. Phys. 155, 489-502 (1993). [16] P. Cartier, Representations of p-adic groups: a survey, Corvallis proceedings, AMS Proc. Sympos. Pure Math., 33 Part. 1, 111-155 (1979). [17] C. Moeglin, M. F. Vignéras, and J. L. Waldspurger, Correspondance de Howe sur un corps p-adique, Lecture Notes in Mathematics 1291, Springer- Verlag, 1987. [18] R. Howe, On the roal of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. , NO. 2. (1980).

SCHOOLOF MATHEMATICS,UNIVERSITYOF SCIENCE AND TECHNOLOGYOF CHINA,HEFEI, 230026, CHINA E-mail address: [email protected], [email protected]