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The Representation Theory of the Heisenberg Group and Beyond

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The Representation Theory of the Heisenberg Group and Beyond The representation theory of the Heisenberg group and beyond Alastair Brodlie ∗) School of Mathematics, University of Leeds In this paper we present some recent and new developments in the theory of p{ mechanics. p{Mechanics is a consistent physical theory which contains both classical and quantum mechanics. The Heisenberg group and its representation theory is the basis of p{mechanics. We give a summary of recent results on p{mechanical observables, states and canonical transformations. In doing so we exhibit relations between the quantum and classical image of these objects. We also present some new work on the Kepler/Coulomb problem. This involves constructing a new Hilbert space which represents the dynamics of the Kepler/Coulomb problem in a simple form. PACS: 03.65.Ca, 03.65.Db. Key words: Heisenberg group, quantum mechanics, coherent states, p{mechanics, canoni- cal transformations, Kepler/Coulomb problem 1 Introduction Since the time of von Neumann the infinite dimensional Schr¨odinger represen- tation of the Heisenberg group has been used in the mathematical formulation of quantum mechanics. By the Stone{von Neumann theorem all unitary irreducible infinite dimensional representations of the Heisenberg group are unitarily equiva- lent to the Schr¨odinger representation. There is also a system of one dimensional representations of the Heisenberg group | these are often ignored and just included in the Stone{von Neumann theorem for mathematical completeness. In [3, 4] it is shown that these one dimensional representations play the same role in classical me- chanics which the infinite dimensional representations play in quantum mechanics. p{Mechanics uses both the one and infinite dimensional representations to obtain relations between classical and quantum mechanics. It shows that both classical and quantum mechanics are derived from the same source separated by the one and infinite dimensional representations respectively. In Section 2 we present some background material on the representation theory of the Heisenberg group. p{Mechanical observables and their time evolution is the subject of Section 3. Section 4 contains a description of the role of states in p{ mechanics. In Section 5 we describe the representation of both linear and non-linear canonical transformations in p{mechanics. In Section 6 we present some new results on the p{mechanical Kepler/Coulomb problem. It is shown that the current form of p{mechanics is insufficient for analysing the Kepler/Coulomb problem. We use Klauder's coherent states for the Kepler/Coulomb problem to derive a new Hilbert space which is particularly useful in studying the Kepler/Coulomb problem. The dynamics for the Kepler/Coulomb problem take a particularly nice form in this ∗) e-mail:[email protected] 1 Alastair Brodlie Hilbert space. This Hilbert space is limited as it can only deal with a subset of observables and states. 2 The Heisenberg group and its representations In this section we present some background material on the Heisenberg group and its representations. Definition 2.1. The Heisenberg group [1, 2] (denoted Hn) is the set of all triples in R × Rn × Rn under the law of multiplication 0 0 0 0 1 0 0 0 0 (s; x; y) · (s ; x ; y ) = s + s + 2 (x · y − x · y); x + x ; y + y : (1) 1 n The non-commutative convolution of two functions B1; B2 2 L (H) is defined as −1 −1 (B1 ∗ B2)(g) = B1(h)B2(h g)dh = B1(gh )B2(h)dh ; Hn Hn Z Z where dh is Harr Measure on Hn, which is just Lebesgue measure on R2n+1, ds dx dy. This operation can be extended to the convolution of two distributions in the natural way | see [2, Chap. 0]. The most common representation of the Heisenberg group is the Schr¨odinger representation. The Schr¨odinger representa- tion [1, Sect. 1.3] for h > 0 is defined on L2(Rn) as S −2πihs+2πixξ+πihxy ρh (s; x; y) (ξ) = e (ξ + hy) : (2) It has been shown that this represen tation is unitary [1, Sect. 1.3], and irreducible [1, Prop. 1.43]. In this paper we only briefly look at this infinite dimensional representation; instead we concentrate on other forms of the infinite dimensional representation and also the often neglected family of one dimensional representa- tions. 2 Definition 2.2. We define the space F (Oh) [3, 4] as 2 2 R2n j F (Oh) = ffh(q; p) 2 L ( ) : Dhfh = 0; for 1 ≤ j ≤ ng ; (3) h @ @ where the operator Dj on L2(R2n) is defined as + i + 2π(p + iq ). h 2 @p @q j j j j 2 An inner product on F (Oh) is given by 4 n 2 hv1; v2iF (Oh) = v1(q; p)v2(q; p) dq dp : (4) h R2n Z 2 F (Oh) is a Hilbert space with respect to this inner product [5, Sect. 4.1]. The n 2 representation ρh [3, 4] of H on F (Oh) is defined by −2πi(hs+qx+py) h h ρh(s; x; y) : fh(q; p) 7! e fh q − 2 y; p + 2 x ; (5) which is unitary with respect to the inner product defined in (4). The crucial theorem which motivates the whole of p{mechanics is 2 The representation theory of the Heisenberg group and beyond Theorem 2.3. (The Stone{von Neumann Theorem) All unitary irreducible repre- sentations of the Heisenberg group, Hn, up to unitary equivalence, are either: 2 (i) of the form ρh on F (Oh) from equation (5), or (ii) for (q; p) 2 R2n the commutative one{dimensional representations on C = 2 L (O(q;p)) −2πi(q·x+p·y) ρ(q;p)(s; x; y)u = e u : (5) For a proof of this theorem see [6]. 3 Observables and their dynamics in p−mechanics The basic idea of p{mechanics is to choose particular functions or distribu- tions on Hn which under the infinite dimensional representation will give quantum mechanical observables, while under the one dimensional representation will give classical mechanical observables. In doing this it is shown that both mechanics are derived from the same source. p{Mechanical observables can be realized as operators (some of which are unbounded) on a subset of L2(Hn) generated by convolutions of the chosen functions or distributions. To define p{mechanical ob- servables properly we need to introduce a map from the set of classical observables to the set of p{mechanical observables. In [3,4] a map of p{mechanization, P, from the set of classical observables to the set of p-mechanical observables is defined as (Pf)(s; x; y) = δ(s)f˘(x; y) ; (6) where f is any classical observable and f˘ is the inverse Fourier transform of f (that ˘ 2πi(qx+py) is, f(x; y) = R2n f(q; p)e dq dp). Definition 3.1R (p−Mechanical Observables). The set of p{mechanical observ- ables is the image of the set of classical observables under the map P from equation (6). Clearly this definition depends on how the set of classical observables is defined. Any physically reasonable classical mechanical observable can be realized as an element of S0(R2n) (see [10] for a definition of S0). Since the Fourier transform maps S0(R2n) into itself, S0(Hn) is a natural choice for the set of p{mechanical observables. It includes the image of all classical observables which are polynomials or exponentials of the variables q and p. If we take the ρh representation (5) of many of the distributions described above we would get unbounded operators. For example the p{mechanisation of @ the classical position coordinate q is the distribution δ(s)δ(x)δ(y); under the @x h @ ρ representation this will become the unbounded operator − 2πiqI. This h 2 @p 2 operator is clearly not defined on the whole of F (Oh). This technical problem can be solved by the usual method of rigged Hilbert spaces (also known as Gelfand triples) [9] which uses the theory of distributions. Another approach to dealing 3 Alastair Brodlie with unbounded operators is given by using the G˚arding space as explained in [1, Chap. 0]. The dynamics of a p{mechanical system are described in [3, 4, 7] using the universal brackets (also known as the p-mechanical brackets). Before we can define the universal brackets we need to define the operator A. @ Definition 3.2. Let S be the operator on C1(Hn). The operator A is defined @s on exponents by 2π e2πihs ; if h 6= 0 ; SA = 4π2I ; where Ae2πihs = ih (7) 8 2 < 4π s ; if h = 0 ; and can be extended by linearity to the whole of:L1(Hn). It can also be extended by linearity to the whole of S(Hn) and then the adjoint is an operator on S0(Hn). A @ is called the anti{derivative operator since it is a right inverse to . @s Definition 3.3. The universal brackets (also known as p{mechanical brackets) are f[B1; B2]g = (B1 ∗ B2 − B2 ∗ B1)A : (8) It is shown in [7, Prop. 3.5] that under the one and infinite dimensional rep- resentations the universal brackets become the Poisson brackets and the quan- tum commutator respectively. Hence for a system with Hamiltonian BH (the p{ mechanisation of the classical Hamiltonian H) solving the p{dynamic equation dB = f[B; B ]g (9) dt H will give the quantum and classical dynamics under the infinite and one dimensional representations respectively. Example 3.4 (The Harmonic Oscillator). The classical Hamiltonian of the harmonic oscillator with frequency ! and mass m is 1 1 H(q; p) = m!2q2 + p2 : (10) 2 m This is a C1 function which can be realized as an element of S0(R2n).
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