Special Functions and Their Symmetries

Vladimir V. Kisil

Course Outline

Chapter 1
Introduction

In the previous part of this course we learn the basic theory of special functions. Some principal elements of the theory are:

1. Integral representations ;

2. Recurrence formulae ;

3. Addition theorems ;

4. Differential equations ;

The rôle of groups and symmetries in science is everlasting, see for example book [32] for inspiring reading. Particularly they make a bridge between geometry and analysis .

Due to large amount of material we chose to adopt the style of books [17,18] which put many facts and examples into exercises for reader.

Chapter 2
Groups, Homogeneous Spaces, and Their Representations

Chapter 3
Groups and Homogeneous Spaces

The group theory and the representation theory are two enormous and interesting subjects themselves. However they are auxiliary in our consideration and we are forced to restrict ourselves only to brief and very dry overview.

Besides introduction to that areas presented in [26,31] we recommend additionally the books [17,30]. The representation theory intensively uses tools of functional analysis and on the other hand inspires its future development. We use the book [18] for references on functional analysis here and recommend it as a nice reading too.

3.1  Basics of Group Theory

Definition 1 A transformation group G is a nonvoid set of mappings of a certain set X into itself with the following properties:

1. if g₁ ∈ G and g₂ ∈ G then g₁g₂ ∈ G;

2. if g ∈ G then g⁻¹ exists and belongs to G.

Exercise 2 List all transformation groups on a set of three elements.

Exercise 3 Verify that the following are groups in fact:

1. Group of permutations of n elements;

2. Group of n×n matrixes with non zero determinant over a field F under matrix multiplications;

3. Group of rotations of the unit circle T;

4. Groups of shifts of the real line R and plane R²;

5. Group of linear fractional transformations of the extended complex plane.

Definition 4 An abstract group (or simply group) is a nonvoid set G on which there is a law of group multiplication (i.e. mapping G ×G→ G) with the properties

1. associativity: g₁(g₂g₃)=(g₁g₂)g₃;

2. the existence of identity: e ∈ G such that eg=ge=g for all g ∈ G;

3. the existence of inverse: for every g ∈ G there exists g⁻¹ ∈ G such that g g⁻¹=g⁻¹g=e.

Exercise 5 Check that any transformation group is an abstract group.

Exercise 6 Check that the following transformation groups (cf. Example 3.1.3) have the same law of multiplication, i.e. are equivalent as abstract groups:

1. The group of isometric mapping of an equilateral triangle onto itself;

2. The group of all permutations of a set of free elements;

3. The group of invertible matrix of order 2 with coefficients in the field of integers modulo 2;

4. The group of linear fractional transformations of the extended complex plane generated by the mappings z→ z⁻¹ and z→ 1−z.

Exercise* 7 Expand the list in the above exercise.

It is simpler to study groups with the following additional property.

Definition 8 A group G is commutative if for all g₁, g₂ ∈ G, we have g₁g₂=g₂g₁.

Most of interesting and important groups are noncommutative , however.

Exercise 9

1. Which groups among found in Exercise 3.1.2 are commutative?

2. Which groups among listed in Exercise 3.1.3 are noncommutative?

Groups could have some additional analytical structures, e.g. they could be a topological sets with a corresponding notion of limit . We always assume that our groups are locally compact  [17,§ 2.4].

Definition 10 If for a group G the group multiplication and the taking of inverse are continuous mappings then G is continuous group .

Even a better structure could be found among Lie groups  [17,§ 6], e.g. groups with a differentiable law of multiplication. Investigating such groups we could employ the whole arsenal of analytical tools, thereafter most of groups studied in this notes will be Lie groups.

Exercise 11 Check that the following are noncommutative Lie (and thus continuous) groups:

1. [30,Chap. 7] The ax+b group : set of elements (a,b), a ∈ R₊, b ∈ R with the group law:
   

   (a, b) * (a′, b′) = (aa′, ab′+b).

   The identity is (1,0), and (a,b)⁻¹=(a⁻¹,−b/a).

2. The Heisenberg group  [15], [30,Chap. 1]: a set of triples of real numbers (s,x,y) with the group multiplication:
   

   (s,x,y)*(s′,x′,y′)=(s+s′+  1 2 (x′y−xy′),x+x′,y+y′).

   (3.1)

   The identity is (0,0,0), and (s,x,y)⁻¹=(−s,−x,−y).

3. The SL₂(R) group [16,23]: a set of 2×2 matrixes with real entries a, b, c, d ∈ R, the determinant det =ad−bc equal to 1 and the group law coinciding with matrix multiplication:
   

      a b c d       a′ b′ c′ d′    =    aa′+bc′ ab′+bd′ ca′+dc′ cb′+dd′    .

   The identity is the unit matrix and
   

      a b c d    −1   =    d −b −c a    .

The above three groups are behind many important results of real and complex analysis [15,16,23] and we meet them many times in these notes.

3.2  Homogeneous Spaces and Invariant Measures

While abstract group are a suitable language for investigation of their general properties we meet groups in applications as transformation groups acting on a set X.

Let X be a set and let be defined an operation G: X→ X of G on X. There is an equivalence relation on X, say, x₁ ∼ x₂ ⇔ ∃g ∈ G: gx₁=x₂, with respect to which X is a disjoint union of distinct orbits [22,§ I.5].

Exercise 1 Let action of SL₂(R) group on C by means of linear-fractional transformations :

   a b c d    : z →  az+b cz+d .

Show that there three orbits: the real axis R, upper (lower) half plane Rⁿ_±:

Rn±={ x±iy   |  x,y ∈ R, y > 0}.

Thus from now on, without lost of a generality, we assume that the operation of G on X is transitive , i. e. for every x ∈ X we have

Gx:=\mathrel ∪ g ∈ G   g(x)=X.

In this case X is G- homogeneous space.

Exercise 2 Show that for any group G we could define its action on X=G as follows:

1. The conjugation g: x → g x g⁻¹ (which is even a group homomorphism, but is trivial for all commutative groups).

2. The left shift λ(g): x → g x and the right shift ρ(g): x → x g⁻¹.

If we fix a point x ∈ X then the set of elements G_x={g ∈ G  |  g(x)=x} obviously forms the isotropy (sub)group of x in G [22,§ I.5]. The set X is in the bijection with the factor set G/G_x for any x ∈ X.

Exercise 3 Find a subgroup which correspond to the given action of G on X:

1. Action of ax+b group on R by the formula: (a,b): x → ax+b.

2. Action of SL₂(R) group on one of three orbit from Exercise 3.2.1.

To do some analysis on groups we need suitably defined basic operation: differentiation and integration . The first operation is naturally defined for Lie group. If G is a Lie group then the homogeneous space G/G_x is a smooth manifold (and a loop as an algebraic object) for every x ∈ X. Therefore the one-to-one mapping G/G_x → X: g→ g(x) induces a structure of C^∞-manifold on S. Thus the class C₀^∞(X) of smooth functions with compact supports on x has the evident definition.

In order to perform an integration we need a suitable measure. A smooth measure dμ on X is called (left) invariant measure with respect to an operation of G on X if

⌠ ⌡ X  f(x) dμ(x) = ⌠ ⌡ X  f(g(x)) dμ(x),    for all g ∈ G,  f(x) ∈ C0∞(X).

(3.2)

Exercise 4 Show that measure y⁻²dy dx on the upper half plane R²₊ is invariant under action from Exercise 3.2.1.

Left invariant measures on X=G is called the Haar measure . It always exists and is uniquely defined up to a scalar multiplier [30,§ 0.2]. An equivalent formulation of (3.2) is: G operates on L₂(X,dμ) by unitary operators. We will transfer the Haar measure dμ from G to \mathfrakg via the exponential map exp:\mathfrakg→ G and will call it as the invariant measure on a Lie algebra \mathfrakg.

Exercise 5 Check that the following are Haar measures for corresponding groups:

1. The Lebesgue measure dx on the real line R.

2. The Lebesgue measure dφ on the unit circle T.

3. dx/x is a Haar measure on the multiplicative group R₊;

4. dx dy/(x²+y²) is a Haar measure on the multiplicative group C\{0}, with coordinates z=x+iy.

5. a⁻² da db and a⁻¹ da db are the left and right invariant measure on ax+b group .

6. The Lebesgue measure ds dx dy of R³ for the Heisenberg group H¹.

In this notes we assume all integrations on groups performed over the Haar measures.

Exercise 6 Show that invariant measure on a compact group G is finite and thus may be normalized to total measure 1.

The above simple result has surprisingly important consequences .

Definition 7 The left convolution f₁*f₂ of two functions f₁(g) and f₂(g) defined on a group G is

f1*f2(g)= ⌠ ⌡ G  f1(h) f2(h−1g) dh

Exercise 8 Let k(g) ∈ L₁(G,dμ) and operator K on L₁(G,dμ) is the left convolution operator with k, .i.e. K: f → k*f. Show that K commutes with all right shifts on G.

The following Lemma characterizes linear subspaces of L₂(G,dμ) invariant under shifts in the term of ideals of convolution algebra L₂(G,dμ) and is of the separate interest.

Lemma 9 A closed linear subspace H of L₂(G,dμ) is invariant under left (right) shifts if and only if H is a left (right) ideal of the right group convolution algebra L₂(G,dμ).

A closed linear subspace H of L₂(G,dμ) is invariant under left (right) shifts if and only if H is a right (left) ideal of the left group convolution algebra L₂(G,dμ).

PROOF. Of course we consider only the ``right-invariance and right-convolution'' case. Then the other three cases are analogous. Let H be a closed linear subspace of L₂(G,dμ) invariant under right shifts and k(g) ∈ H. We will show the inclusion

[f*k]r(h)= ⌠ ⌡ G  f(g)k(hg) dμ(g) ∈ H,

(3.3)

for any f ∈ L₂(G,dμ). Indeed, we can treat integral (3.3) as a limit of sums

N ∑ j=1  f(gj)k(hgj)∆j.

(3.4)

But the last sum is simply a linear combination of vectors k(hg_j) ∈ H (by the invariance of H) with coefficients f(g_j). Therefore sum (3.4) belongs to H and this is true for integral (3.3) by the closeness of H.

Otherwise, let H be a right ideal in the group convolution algebra L₂(G,dμ) and let φ_j(g) ∈ L₂(G,dμ) be an approximate unit of the algebra [12,§ 13.2], i. e. for any f ∈ L₂(G,dμ) we have

[φj*f]r(h)= ⌠ ⌡ G  φj(g)f(hg) dμ(g) → f(h), when j→∞.

Then for k(g) ∈ H and for any h′ ∈ G the right convolution

[φj*k]r(hh′)= ⌠ ⌡ G  φj(g)k(hh′g) dμ(g) = ⌠ ⌡ G  φj(h′−1g′)k(hg′) dμ(g′), g′=h′g,

from the first expression is tensing to k(hh′) and from the second one belongs to H (as a right ideal). Again the closeness of H implies k(hh′) ∈ H that proves the assertion. ^[¯]

Chapter 4
Elements of the Representation Theory

4.1  Representations of Groups

Definition 1 A linear continuous representation of a group G is a continuous function T(g) on G with values in the group of non-degenerate linear continuous transformation in a linear space H (either finite or infinite dimensional) such that T(g) satisfies to the functional identity:

T(g1 g2) = T(g1) T(g2).

(4.1)

Exercise 2 Show that T(g⁻¹)=T⁻¹(g) and T(e)=I, where I is the identity operator on B.

Exercise 3 Show that these are linear continuous representations of corresponding groups:

1. Operators T(x) such that [T(x) f](t)=f(t+x) form a representation of R in L₂(R).

2. Operators T(n) such that T(n) a_k=a_k+n form a representation of Z in l₂.

3. Operators T(a,b) defined by
   

   [T(a,b) f](x) = √af(ax+b),        a ∈ R+,  b ∈ R

   (4.2)

   form a representation of ax+b group in L₂(R).

4. Operators T(s,x,y) defined by
   

   [T(s,x,y) f] (t)=ei(2s−√2yt+xy) f(t− √2x)

   (4.3)

   form Schrödinger representation of the Heisenberg group H¹ in L₂(R).

5. Operators T(g) defined by
   

   [T(g) f](t) =  1 ct+d f    at+b ct+d   ,     where g=    a b c d    ,

   (4.4)

   form a representation of SL₂(R) in L₂(R).

In the sequel a representation always means linear continuous representation. T(g) is an exact representation (or faithful representation if T(g)=I only for g=e. The opposite case when T(g)=I for all g ∈ G is a trivial representation . The space H is representation space and in most cases will be a Hilber space [18,§ III.5]. If dimensionality of H is finite then T is a finite dimensional representation , in the opposite case it is infinite dimensional representation .

We denote the scalar product on H by 〈 ·,· 〉. Let {e_j} be an (finite or infinite) orthonormal basis in H, i.e.

〈 ej,ej 〉=δjk,

where δ_jk is the Kroneker delta, and linear span of {e_j} is dense in H.

Definition 4 The matrix elements t_jk(g) of a representation T of a group G (with respect to a basis {e_j} in H) are complex valued functions on G defined by

tjk(g) = 〈 T(g)ej,ek 〉.

(4.5)

Exercise 5 Show that [31,§ 1.1.3]

1. T(g) e_k=∑_j t_jk(g) e_j.

2. t_jk(g₁g₂)=∑_n t_jn(g₁) t_nk(g₂).

It is typical mathematical questions to determine identical objects which may have a different appearance. For representations it is solved in the following definition.

Definition 6 Two representations T₁ and T₂ of the same group G in spaces H₁ and H₂ correspondingly are equivalent representations if there exist a linear operator A: H₁ → H₂ with the continuous inverse operator A⁻¹ such that:

T2(g) = A T1(g) A−1,        ∀g ∈ G.

Exercise 7 Show that representation T(a,b) of ax+b group in L₂(R) from Exercise 3 is equivalent to the representation

[T1(a,b) f] (x) =  ei[ b/a] √a   f    x a   .

(4.6)

HINT. Use the Fourier transform. ^[¯]

The relation of equivalence is reflexive, symmetric, and transitive. Thus it splits the set of all representations of a group G into classes of equivalent representations. In the sequel we study group representations up to their equivalence classes only.

Exercise 8 Show that equivalent representations have the same matrix elements in appropriate basis.

Definition 9 Let T is a representation of a group G in H The adjoint representation T′(g) of G in H is defined by

T′(g)=( T(g−1))*,

where ^* denotes the adjoint operator in H.

Exercise 10 Show that

1. T′ is indeed a representation.

2. t′_jk(g)=t_kj(g⁻¹).

Recall [18,§ III.5.2] that a bijection U: H → H is a unitary operator if

〈 Ux,Uy 〉=〈 x,y 〉,        ∀x, y ∈ H.

Exercise 11 Show that UU^*=I.

Definition 12 T is a unitary representation of a group G in a space H if T(g) is a unitary operator for all g ∈ G. T₁ and T₂ are unitary equivalent representations if T₂=UT₂U⁻¹ for a unitary operator U.

Exercise 13

1. Show that all representations from Exercises 4.1.3 are unitary.

2. Show that representations from Exercises 3 and 4.1.7 are unitary equivalent.

HINT. Take that the Fourier transform is unitary for granted. ^[¯]

Exercise 14 Show that if a Lie group G is represented by unitary operators in H then its Lie algebra \mathfrakg is represented by self-adjoint (possibly unbounded) operators in H.

The following definition have a sense for finite dimensional representations.

Definition 15 A character of representation T is equal χ(g) = tr (T(g)), where tr  is the trace [18,§ III.5.2 (Probl.)] of operator.

Exercise 16 Show that

1. Characters of a representation T are constant on the adjoint elements g⁻¹hg, for all g ∈ G.

2. Character is an algebra homomorphism from an algebra of representations with Kronecker's (tensor) multiplication [31,§ 1.9] to complex numbers.

HINT. Use that tr (AB)=tr (BA), tr (A+B)=tr A + tr B, and tr ( A ⊗B) = tr A tr B. ^[¯]

For infinite dimensional representation characters could be defined either as distributions [17,§ 11.2] or in infinitesimal terms of Lie algebras [17,§ 11.3].

The characters of a representation should not be confused with the following notion.

Definition 17 A character of a group G is a one-dimensional representation of G.

Exercise 18

1. Let χ be a character of a group G. Show that a character of representation χ coincides with it and thus is a character of G.

2. A matrix element of a group character χ coincides with χ.

3. Let χ₁ and χ₁ be characters of a group G. Show that χ₁ ⊗χ₂ = χ₁χ₂ and χ′(g)=χ₁(g⁻¹) are again characters of G. In other words characters of a group form a group themselves.

4.2  Decomposition of Representations

1. The main theorem of arithmetics on unique representation an integer as a product of powers of prime numbers.

2. Jordan's normal form of a matrix.

Definition 1 Let T be a representation of G in H. A linear subspace L ⊂ H is invariant subspace for T if for any x ∈ L and any g ∈ G the vector T(g)x again belong to L.

There are always two trivial invariant subspaces: the null and entire H. All other are nontrivial invariant subspaces.

Definition 2 If there are only two trivial invariant subspaces then T is irreducible representation . In the opposite case we have reducible representation .

For any nontrivial invariant subspace we could define the restriction of representation of T on it. In this way we obtain a subrepresentation of T.

Example 3 Let T(a), a ∈ R₊ be defined as follows: [T(a)]f(x)=f(ax). Then spaces of even and odd functions are invariant.

Definition 4 If the closure of liner span of all vectors T(g) v is dense in H then v is called cyclic vector for T.

Exercise 5 Show that for an irreducible representation any non zero vector is cyclic.

The important property of unitary representation is complete reducibility.

Exercise 6 Let a unitary representation T has an invariant subspace L ⊂ H, then its orthogonal completion L^⊥ is also invariant.

Theorem 7 [17,§ 8.4] Any unitary representation T of a locally compact group G could be decomposed in a (continuous) direct sum irreducible representations: T=∫_X T_x  dμ(x).

The necessity of continuous sums appeared in very simple examples:

Exercise 8 Let T be a representation of R in L₂(R) as follows: [T(a)f](x)=e^iaxf(x). Show that

1. Any measurable set E ⊂ R define an invariant subspace of functions vanishing outside E.

2. T does not have invariant irreducible subrepresentations.

Definition 9 The set of equivalence classes of unitary irreducible representations of a group G is denoted be ∧G and called dual object (or dual space) of the group G.

Definition 10 A left regular representation Λ(g) of a group G is the representation by left shifts in the space L₂(G) of square-integrable function on G with the left Haar measure

Λg: f(h) → f(g−1h).

(4.7)

The main problem of representation theory is to decompose a left regular representation Λ(g) into irreducible components.

4.3  Invariant Operators and Schur's Lemma

It is a pleasant feature of an abstract theory that we obtain important general statements from simple observations. Finiteness of invariant measure on a compact group is one such example. Another example is Schur's Lemma presented here.

To find different classes of representations we need to compare them each other. This is done by intertwining operators.

Definition 1 Let T₁ and T₂ are representations of a group G in a spaces H₁ and H₂ correspondingly. An operator A: H₁ → H₂ is called an intertwining operator if

A T1(g) = T2(g) A,        ∀g ∈ G.

If T₁=T₂=T then A is interntwinig operator or commuting operator for T.

Exercise 2 Let G, H, T(g), and A be as above. Show that [31,§ 1.3.1]

1. Let x ∈ H be an eigenvector for A with eigenvalue λ. Then T(g)x for all g ∈ G are eigenvectors of A with the same eigenvalue λ.

2. All eigenvectors of A with a fixed eigenvalue λ for a linear subspace invariant under all T(g), g ∈ G.

3. If an operator A is commuting with irreducible representation T then A=λI.

HINT. Use the spectral decomposition of selfadjoint operators [18,§ V.2.2]. ^[¯]

The next result have very important applications.

Lemma 3 [Schur] [17,§ 8.2] If two representations T₁ and T₂ of a group G are irreducible, then every intertwining operator between them either zero or is invertible.

HINT. Consider subspaces kerA ⊂ H₁ and im A ⊂ H₂. ^[¯]

Exercise 4 Show that

1. Two irreducible representations either equivalent or disjunctive.

2. All operators commuting with an irreducible representation form a field.

3. Irreducible representation of commutative group are one-dimensional.

4. If T is unitary irreducible representation in H and B(·,·) is a bounded semi linear form in H invariant under T: B(T(g)x,T(g)y)=B(x,y) then B(·,·)=λ〈 ·,· 〉.

HINT. Use that B(·,·)=〈 A·,· 〉 for some A [18,§ III.5.1]. ^[¯]

Chapter 5
Group of Reals and Harmonic Analysis

As we know the exponential and trigonometric functions are special (hypergeometric ) functions. Thus they will be our first examples of special functions arisen from representations of group according to the following definition.

Definition 1 A special function associated with a representation T of a group G is a matrix element t_ij(g) of T.

We recall that matrix elements of group characters coincide with themselves .

5.1  Exponential and Trigonometric Functions

The group of translations of the real line R is commutative thus by Exercise 3 all its irreducible representations are one dimensional, i.e. characters -functions R → C which satisfy to the functional equation :

f(x) f(y)=f(x+y).

Recall that characters form a group .

Exercise 1 Show that from the above equation follow that

1. f(x) should be differentiable infinitely many times;

2. f(x) satisfies to f′(0)f(x)=f′(x) and thus is f(x)=e^ax with a=f′(0).

Theorem 2 All unitary irreducible representations of R are T(x)=e^iax for an arbitrary a ∈ R. In other words ∧R=R.

Exercise 3 Show that all irreducible representations of the multiplicative group R₊ are of the form T(t)=t^a for arbitrary a ∈ C. They are unitary if a=ib, b ∈ R.

HINT. Use the group homomorphism exp: R → R₊. ^[¯]

The group of rotations SO(2) of a unit circle T or Euclidean plane preserving quadratic form x²+y² is also commutative. A rotation by an angle φ described in Cartesian coordinates by the matrix:

g(φ)=    cosφ −sinφ sinφ cosφ    ,

(5.1)

which could be considered as definition of trigonometric functions sin and cos. From the functional identity g(φ) g(ψ) = g(φ+ψ) translated to the matrix multiplication follows addition formulae :

sin(α±β) = sinαcosβ±sinβ cosα cos(α±β) = cosαcosβ±sin αsinβ

This is the first occurrence of important formulae which we oftenly meet later. In general case addition formulae are realization of the property of matrix elements 2.

To find all irreducible unitary representations of SO(2) we will use that SO(2)=R/Z_2π.

Exercise 4 Show that all irreducible unitary representations of SO(2) hev the form g(φ)=e^inφ, n ∈ Z. In other words, ∧T=Z.

Exercise 5 Decompose representation (5.1) into irreducible components.

HINT. Use the unitary transformation:

1 2    1 i i 1       cosφ −sinφ sinφ cosφ       1 −i −i 1    =    eiφ 0 0 e−iφ   

^[¯]

Exercise* 6 Find

1. All unitary irreducible representation of the group SH(2) of hyperbolic rotations, i.e. preserving the quadratic form x²−y² on R².

2. Corresponding addition formulae.

5.2  Duality and the Fourier Transform

Exercise 1

1. Show that ∧Z=T.

2. Collect the above result together with Theorem 5.1.2 and Exercise 5.1.4 to obtain that dual object of the Abelian group Rⁿ×Z^k×T^l is the group Rⁿ×T^k×Z^l.

The above result is a particular case of the Pontrjagin's duality, which plays an exceptional rôle in the representation theory of Abelian groups

Theorem 2 [Pontrjagin's duality] [17,§ 12.1], [18,§ IV.2.1] For an arbitrary locally compact abelian ( l.c.a. ) group G, the canonical mapping of G into ∧∧G is an isomorphism of topological groups. Haar measures on G=∧∧G and ∧G can be normalized so that :

^ f   (χ) = ⌠ ⌡ G  f(g) χ(g) dg (5.2) f(g) = ⌠ ⌡ ∧G  ^ f   (χ)  χ(g)    dχ (5.3) ⌠ ⌡ G  | f(g) |2 dg = ⌠ ⌡ ∧G    ^ f   (χ)   2    dχ (5.4)

Particularly normalized to 1 measure on a compact group corresponds to a point measure 1 on the discrete dual.

The formulas above deserve special names.

Definition 3 The transformations (5.2) is called the Fourier transform from G to ∧G, (5.3) is inverse Fourier transform from ∧G to G=∧∧G, and (5.4) is known as Plancherel's identity.

The following is remarkable properties of the Fourier transform.

Theorem 4 Let G is a l.c.a. group with an invariant measure μ. The Fourier transform maps

1. L₁(G,μ) into the space of continuous bounded functions on ∧G.

2. Convolutions into multiplication: (f₁*f₂)∧ (χ)=∧f₁(χ)·∧f₂(χ).

3. Shifts into multiplication by the character: (λ(g)f)∧ (χ)=χ(g)∧f(χ).

4. Multiplication by a character χ₁ ∈ ∧G to the shifts on ∧G: (χ₁· f)∧ (χ)=∧f(χ⁻¹₁χ).

5.3  Fourier Series

We consider T=SO(2) which is a simplest example of compact group . We state here without proofs main features of representation theory of compact groups.

Theorem 1 [17,§ 9.2]

1. Every topologically irreducible representation of a compact group G is finite-dimensional and unitarizable.

2. If T₁ and T₂ are two inequivalent irreducible representations, then every matrix element of T₁ is orthogonal in L₂(G) to every matrix element of T₂.

3. For a compact group G its dual space ∧G is discrete.

Due to relation SO(2)=R/Z_2π we can identify functions on T with periodic functions on R with a period 2π. We define invariant integration on SO(()2) by the invariant measure from Exercise 2:

⌠ ⌡ SO(2)  f(g) dg=  1 2π ⌠ ⌡ 2π 0  f(φ) dφ.

Theorem 2 Functions {e^inφ} form a complete orthonormal system in L₂(SO(2)):

1 2π ⌠ ⌡ 2π 0  eimφ  einφ    dφ = δnm.

Therefore any function f(φ) ∈ L₂(SO(2)) can be represented by its Fourier series (cf. (5.3)):

f(φ)= ∞ ∑ n=−∞  cn einφ,

where Fourier coefficients c_n defined from (cf. (5.2))

cn=〈 f(φ),einφ 〉 =  1 2π ⌠ ⌡ 2π 0  f(φ)  einφ    dφ

Corollary 3 [Plancherel Identity] (cf. (5.4))

1 2π ⌠ ⌡ 2π 0  | f(φ) |2 dφ = ∞ ∑ n=−∞  cn2.

The main problem of representation theory for SO(2) has the following solution:

Theorem 4 The regular representation R(φ) of G is a direct sum of one dimensional representations T_n(φ)=e^inφ with the multiplicity 1:

R(φ)= ∞ ∑ n=−∞  Tn(φ).

Exercise 5

1. Find the particular form of the general Theorem 5.2.4 for G=SO(2).

2. The Fourier transform on SO(2) sends the derivative to the operator of multiplication by the sequence {2π in}_{n ∈ Z}.

As a corollary we could advance to the point-wise convergence of the Fourier series:

Exercise 6 If f(φ) on SO(2) is differentiable infinitely many times

1. then its Fourier coefficients decrease rapidly:
   

   lim n→ ∞  nk cn=0,        ∀k ∈ N.

2. then its Fourier series converges point-wise.

Exercise 7 If Fourier coefficients of a function f(φ) decrease rapidly then it is differentiable infinitely many time.

5.4  Fourier Integral

The decomposition of the left regular representation into irreducible components combines the Fourier transform and decomposition in Example 4.2.8.

Exercise 1

1. Find the particular form of the general Theorem 5.2.4 for G=R.

2. The Fourier transform on R sends the derivative [ d/dx] to the operator of multiplication by 2πiξ.

It is turn to be that commutative harmonic analysis could be better understood with help of representation theory of the non-commutative Heisenberg group H¹ [15].

Exercise 2 Check that

1. ρ defined by
   

   ρ(s,x,y) f(t) = ei(s+xt+xy/2)f(t+y)

   (5.5)

   is a unitary representation of H¹ in L₂(R).

2. ρ is irreducible (Hint. It may commute only with operators of multiplication by a constant function).

3. r(s,x,y)=r(s,−y,x) is an automorphism of H¹.

4. The Fourier transform intertwines two representations ρ and ρ°r of H¹: ∧  ρ(h)=ρ(r(h))∧ .

From the above Exercise and the Schur's Lemma we conclude that the Fourier transform is a unitary operator up to a scalar multiplier. By iteration we know that H¹: ∧ ∧  ρ(h)=ρ(r²(h))∧ ∧  and r²(s,x,y)=r(s,−x,−y). From that ∧ ²=w(−1), where w(−1)f(t)=f(−t).

Lemma 3 Show that the scalar multiplier is equal to 1 by demonstration that (e^−πx2)∧ =e^−πx2. (This formula were stated in the first part of the course).

PROOF. The function e^−πx2 solves the equation (d/dx +2πx)f=0 which is invariant under the Fourier transform due to Exercise 2. Then e^−πx2 is the Fourier transform up to a scalar factor, which is equal to 1 from the formula:

⌠ ⌡ ∞ −∞  e−πx2 dx=1.

(It was proven in the first part of the course). ^[¯]

Remark 4 The above results of harmonic analysis could be extended for general l.c.a. group , particularly for Rⁿ×Z^k×T^l.

Chapter 6
Harmonic Analysis on the Sphere

6.1  Rotations of the Euclidean Space

Let us start from two problems [17,§ 17.1]:

Problem 1 A convex centrally symmetric body in Rⁿ is uniquely determined by the area of its projections on all possible hyperplanes.

Problem 2 A convex centrally symmetric body K in Rⁿ is uniquely determined by the areas of its sections by all possible hyperplanes.

In fact we have two faces of the same problem.

Exercise 3 Show that Problems 6.1.1 and 6.1.2 are equivalent.

HINT. Use a norm in Rⁿ defined by a convex centrally symmetric body. Show that projections and section define dual norms. ^[¯]

We restrict ourself to the case n=3. We could describe a convex body by an even function on S ⊂ R³:

f(x)=  1 2 r2x,       x ∈ S,

where r_x distance from 0 to the boundary in direction of x.

Exercise 4 Let C be a great circle of S on a plane \spaceP. Then area (K∩P)=∫_C f(x) dx.

Exercise 5 Problem 6.1.2 is equivalent to: an even function on the sphere is uniquely determined by its integrals on all great circles.

Let L₂(S) be the space of square integrable function on S and L₂⁺(S) its subspace of the even functions. We have natural representations T and T₊ of the group of isometric rotations SO(3) of Euclidean space R³ in L₂(S) and L₂⁺(S) respectively. We define operator J

Jf(x)= ⌠ ⌡ Cx  f(y) dy,

(6.1)

where C_x is the great circle with epicenter at the point x ∈ S.

Exercise 6 The operator J intertwines T and T₊.

Exercise 7 Problem 6.1.2 is equivalent to ker J=0 in L₂⁺(S).

From Theorem 5.3.1 L₂⁺(S) is a direct sum of irreducible finite dimensional spaces in each of which the operator J is scalar by the Schur's Lemma . Let P_n be the space of all functions on S that are restriction to S of homogeneous polynomials of degree n in R³.

Exercise 8 Prove

1. P_n ⊂ P_n+2 (use x²+y²+z²=1 on S).

2. dimPⁿ=(n+1)(n+2)/2 (use induction).

Let H_n be the orthogonal completion of P_n−2 in P_n.

Theorem 9 The decomposition of the spaces L₂(S) and L₂⁺(S) into irreducible subspaces for T and T₊ of SO(3) have the form:

L2(S)= ∞ ∑ n=0  Hn        and        L2+(S)= ∞ ∑ n=0  H2n,

respectively.

PROOF. Decompositions are valid the standard functional analytical reasoning. The invariance of H_n is also obvious. The remaining part is irreducibility of H_n which is proven by the next two Exercises. ^[¯]

Exercise 10 H_n contains exactly one function L_n that is invariant under the subgroup of rotations about the z-axis.

HINT. Consider [n/2]+1 functions in P_n: zⁿ, zⁿ⁻²(x²+y²), ..., z^n−2[n/2](x²+y²). ^[¯]

Exercise 11 Prove that every irreducible subspace V ⊂ L₂(S) contains at least one non-zero function that is invariant under rotations about the z-axis.

HINT. Use invariant integration . ^[¯]

To conclude solution of Exercise 6.1.7 (and thus Problems 6.1.1 and 6.1.2) we will explicitly calculate eigenvalues of J in H_n

Exercise 12 Show that z-invariant function L_n could be taken to be n-th Legendre polynomial :

Ln(z)=  dn dzn [(z2−1)n].

HINT. Prove for functions f on S that depends only on the coordinate z (use integration in cylindrical coordinates ):

⌠ ⌡ S  f(x) dx=π ⌠ ⌡ 1 −1  f(z) dz.

Use integration by parts to prove that L_n orthogonal to all polynomials in z of degree less than n with respect to the following inner product:

(f1,f2)=π ⌠ ⌡ 1 −1  f1(z)  - f   2  (z) dz.

Show that L_n uniquely defined up to scalar factor by the above properties. ^[¯]

The Legendre polynomials are another example of the special functions , more precisely they are orthogonal polynomials .

Substituting L_n instead of f and (0,0,1) instead of x in (6.1) we have:

λn Ln(1)=2πLn(0).

Obviously:

Ln(1) =  dn dzn [(z−1)n(z+1)n]   z=1  = n!(z+1)n|z=1 = 2n·n!, Ln(0) =      0 if n is odd, (2k)!   2k k   if n=2k is even.

As the result:

λn(0) =      0 if n is odd, 2π  (2k−1)!! 2k!! if n=2k is even,

where n!!=n(n−2)(n−4)…. This finishes the consideration.

Chapter 7
Hermit Polynomials, Heisenberg Group, and Segal-Bargmann Spaces

2

7.1  Introduction

This lecture is based on the paper [8].

It is well known, by the celebrated Stone-von Neumann theorem, that all models for the canonical quantisation [24] are isomorphic and provide us with equivalent representations of the Heisenberg group  [30,Chap. 1]. Nevertheless it is worthwhile to look for some models which can act as alternatives for the Schrödinger representation. In particular, the Segal-Bargmann representation [2,28] serves to

• give a geometric representation of the dynamics of the harmonic oscillators ;

• present a nice model for the creation and annihilation operators, which is important for quantum field theory;

• allow applying tools of analytic function theory .

We look for similar connections between nilpotent Lie groups and spaces of monogenic [5,10] Clifford valued functions. Particularly we are interested in a third possible representation of the Heisenberg group, acting on monogenic functions on Rⁿ. There are several reasons why such a model can be of interest. First of all the theory of monogenic functions is (at least) as interesting as several complex variable theory, so the monogenic model should share many pleasant features with the Segal-Bargmann model. Moreover, monogenic functions take their value in a Clifford algebra, which is a natural environment in which to represent internal degrees of freedom of elementary particles such as spin. Thus from the very beginning it has a structure which in the Segal-Bargmann model has to be added, usually by means of the second quantization procedure [11]. So a monogenic representation can be even more relevant to quantum field theory than the Segal-Bargmann one (see Remark 7.3.2).

From the different aspects of the Segal-Bargmann space F₂( Cⁿ ) we select the one giving a unitary representation of the Heisenberg group Hⁿ. The representation is unitary equivalent to the Schrödinger representation on L₂( Rⁿ ) and the Segal-Bargmann transform is precisely the intertwining operator between these two representations (see subsection 7.3.2).

This lecture is closely related to [21], where connections between analytic function theories and group representations were described. Representations of another group (SL(2,R)) in spaces of monogenic functions can be found in [20]. We hope that the present lecture make only few first steps towards an interesting function theory and other steps will be done elsewhere.

7.2  Wavelets or Coherent States

In our approach we will need some basic facts on wavelets (or coherent states) and associated wavelet transform.

Let G be a group which acts via transformation of a closed domain Ω. Moreover, let G: ∂Ω→ ∂Ω and G act on Ω and ∂Ω transitively. Let us fix a point x₀ ∈ Ω and let H ⊂ G be a stationary subgroup of point x₀. Then domain Ω is naturally identified with the homogeneous space G/H. Till the moment we do not request anything untypical. Now let

• there exist a H-invariant measure dμ on ∂ Ω.

1. f₀ ∈ F₂(∂Ω, dμ);

2. F₂(∂Ω, dμ) is G-invariant;

3. F₂(∂Ω, dμ) is G-irreducible, or f₀ is cyclic in F₂(∂ Ω, dμ).

We define the inverse wavelet transform M according to the formula:

[ M ^ f   ](x) = ⌠ ⌡ Ω  ^ f   (a) fs(a)(x)  da ,

(7.2)

The following proposition explain the usage of the name for M.

Theorem 1 The operator

P = M W: B → B

(7.3)

is a projection of B to its linear subspace for which b₀ is cyclic. Particularly if π is an irreducible representation then the inverse wavelet transform M is a left inverse operator on B for the wavelet transform W:

MW=I.

7.3  The Heisenberg Group and Spaces of Analytic Functions

7.3.1  The Schrödinger Representation of the heisenberg Group

The Lie algebra of the Heisenberg group is generated by the 2n+1 elements p₁, ..., p_n, q₁, ..., q_n, e, with the well-known Heisenberg commutator relations:

[pi,qj]=δij e.

(7.4)

All other commutators vanish. In the standard quantum mechanical interpretation the operators are momentum and coordinate operators [14,§ 1.1].

It is common practice to switch between real and complex Lie algebras. Complexify \mathfrakhⁿ to obtain the complex algebra C\mathfrakhⁿ, and take four complex numbers a, b, c and d such that ad−bc ≠ 0. The real 2n+1-dimensional subspace spanned by

Ak=apk+bqk       Bk=cpk+dqk

and the commutator [A_k,B_k]=(ad−bc)e, where e=[p_k,q_k] is of course isomorphic to \mathfrakhⁿ, and exponentiating will give a group isomorphic to the Heisenberg group.

An example of this procedure is obtained from the construction of the so-called creation and annihilation operators of Bose particles in the k-th state, a⁺_k and a⁻_k (see [14,§ 1.1]). These are defined by:

a±k=  qk ±ιpk √2 ,

(7.5)

giving the commutators [a⁺_i,a⁻_j]=(−ι)δ_ije. Putting −ιe=l, the real algebra spanned by a^±_k and l is an alternative realization of \mathfrakhⁿ, \mathfrakhⁿ_a.

An element g of the Heisenberg group Hⁿ (for any positive integer n, cf. (3.1)) can be represented as g=(t,z) with t ∈ R, z =(z₁,…,z_n) ∈ Cⁿ. The group law in coordinates (t,z) is given by

g*g′=(t,z)*(t′,z′)=(t+t′+  1 2 n ∑ j=1  ℑ( - z   j  zj′), z+z′),

(7.6)

where ℑz denotes the imaginary part of the complex number z. Of course the Heisenberg group is non-commutative.

The relation between the Heisenberg group and its Lie algebra is given by the exponentiation exp:\mathfrakhⁿ_a→Hⁿ. We define the formal vector a⁺ as being (a⁺₁,…,a⁺_n) and a⁻ as (a⁻₁,…,a⁻_n), which allows us to use the formal inner products

u·a+ = n ∑ k=1  uka+k v·a− = n ∑ k=1  vka−k.

With these we define, for real vectors u and v, and real s

exp(u·(a++a−)) = (0,√2u) (7.7) exp(v·(a−−a+) = (0,ιv) (7.8) exp(s l) = (e−2s,0). (7.9)

Possible Schrödinger representations (cf. (5.5)) are parameterized by the non-zero real number (^h/_2p) (the Planck constant). As usual, for considerations where the correspondence principle between classic and quantum mechanics is irrelevant, we consider only the case (^h/_2p) = 1. The Hilbert space for the Schrödinger representation is L₂ (Rⁿ), where elements of the complex Lie algebra C\mathfrakhⁿ are represented by the unbounded operators

σ(a±k) =  1 √2   xk I ±  ∂ ∂xk   .

(7.10)

From which it follows, using any j, that

σ(l)=[a+j,a−j]=−2 I.

The corresponding representation π of the Heisenberg group is given by exponentiation of the σ(a⁺_k) and σ(a⁻_k), but this is most readily expressed by using p_k and q_k, and so is generated by shifts and multiplications s_c: f(x) → f(x+c) and m_b: f(x)→ e^ιx·bf(x), with the Weyl commutation relation

scmb = eιc·b mbsc.

There is an orthonormal basis of L₂(Rⁿ) on which the operators σ(a^±_k) act in an especially simple way. It consists of the functions:

φm(y)=[2m m!    √   π   ]−1/2 e−x·x/2Hm(y),

(7.11)

where y =(y₁,…,y_n), m=(m₁,…,m_n), and H_m(y) is the generalized Hermite polynomial

Hm(y)= n ∏ i=1  Hmi(yi).

For these

a+k φm(y)= √   mk+1    φm′(y),        a−k φm(y) = √   mk    φm′′(y)

where

m′ = (m1,m2,…, mk−1, mk + 1, mk+1,…,mn) m′′ = (m1,m2,…, mk−1, mk − 1, mk+1,…,mn).

This is the most straightforward way to express the creation or annihilation of a particle in the k-th state.

Let us now consider the generating function of the φ_m(x),

A(x,y)= ∞ ∑ j=0   xj √ j! φk(y) = exp(−  1 2 (x·x+y·y) + √2x·y).

(7.12)

We state the following elementary fact in Dirac's bra-ket notation.

Lemma 1 Let H and H′ be two Hilbert spaces with orthonormal bases {φ_k} and {φ′_k} respectively. Then the sum

U= ∞ ∑ j=0  |φ′j〉 〈 φj|

(7.13)

defines a unitary operator U: H → H′ with the following properties:

1. U φ_k = φ_k′;

2. If an operator T: H→ H is expressed, relative to the basis φ_k, by the matrix (a_ij) then the operator UTU⁻¹: H′→ H′ is expressed relative to the basis φ_k′ by the same matrix.

Now, if we take the function A(x,y) from (7.12) as a kernel for an integral transform ,

[Af](y)= ⌠ ⌡ Rn  A(y,x) f(x) dx

we can consider it subject to the Lemma above. However, for this we need to define the space H′ and an orthonormal basis {φ_k′} (we already identified H with L₂(Rⁿ) and the {φ_k} are given by (7.11)). There is some freedom in doing this.

For example it is possible to take the holomorphic extension A(z,y) of A(x,y) with respect to the first variable. Then

1. H′ is the Segal-Bargmann space of analytic functions over Cⁿ with scalar product defined by the integral with respect to Gaussian measure e^{−| z|2} dz;

2. The Heisenberg group acts on the Segal-Bargmann space as follows:
   

   [β(t,z)f](u)=f(u+z)eιt−z·u−| z|2/2.

   (7.14)

   This action generates the set of coherent states f_(0,v)(u)=e^{−vu−| v|2/2}, u, v ∈ Cⁿ from the vacuum vector f₀(u) ≡ 1;

3. The operators of creation and annihilation are a⁺_k=z_k I, a⁻_k=[ ∂/(∂z_k)].

4. The Segal-Bargmann space is spanned by the orthonormal basis φ_k′=[ 1/√{m!}]zⁿ or by the set of coherent states f_(0,v)(u)=e^{−vu−| v|2/2}, u, v ∈ Cⁿ

5. The intertwining kernel for σ_(t,z) (7.10) and β_(t,z) (7.14) is
   

   A(z,y)=e−(z·z+x·x)/2−√2z·x = ∞ ∑ k=0   zm √ m! ·  1 √ 2m m! 4 √ π e−x·x/2Hm(y)

6. The Segal-Bargmann space has a reproducing kernel
   

   K(u,v) = eu·v= ∞ ∑ k=1  φk(u) - φ   k  (v) = ⌠ ⌡ eu·z ez·v e−| z|2 dz.

7.3.2  The Segal-Bargmann space

As ``vacuum vector'' we will select the original vacuum vector of quantum mechanics-the Gauss function f₀(x)=e^−x·x/2. Its transformations are defined as follows:

wg(x)=π(t,z) f0(x) = eι(2t−√2q·x+q·p) e−(x− √2p)2/2 = e2ιt−(p·p+q·q)/2e− ((p−ιq)2+x·x)/2+√2(p−ιq)·x = e2ιt−z·z/2e− (z·z+x·x)/2+√2z·x.

In particular w_(t,0)(x)=e^−2itf₀(x), i.e. it really is a vacuum vector with respect to ~H in the sense of our definition. Of course ~G / ~H is isomorphic to Cⁿ. Embedding Cⁿ in G by the identification of (0,z) with z, the mapping s:~G → ~G is defined simply by s((t,z))=(0,z)=z; Ω then is identical with Cⁿ.

The Haar measure on Hⁿ coincides with the standard Lebesgue measure on R²ⁿ⁺¹  [30,§ 1.1] and so the invariant measure on Ω also coincides with Lebesgue measure on Cⁿ. Note also that the composition law sending z₁ z₂ to s((0,z₁)(0,z₂)) reduces to Euclidean shifts on Cⁿ . We also find s((0,z₁)⁻¹·(0,z₂))=z₂−z₁ and r((0,z₁)⁻¹·(0,z₂)) = ([ 1/2] ℑz₁·z₂,0).

The reduced wavelet transform takes the form of a mapping L₂(Rⁿ ) → L₂( Cⁿ ) and is given by the formula

^ W   f(z) = 〈 f,w(0,z) 〉 = π−n/4 ⌠ ⌡ Rn  f(x) e−z·z/2 e−(z·z+x·x)/2+√2z·x dx = e−| z|2/2π−n/4 ⌠ ⌡ Rn  f(x) e−(z·z+x·x)/2+√2z·x dx, (7.16)

where z =p+ιq. Then ∧Wf belongs to L₂( Cⁿ , dg). This can better be expressed by saying that the function \brevef(z)=e^{| z|2/2}∧Wf(z) belongs to L₂( Cⁿ , e^{− | z|2} dg) because \brevef(z) is analytic in z. These functions constitute the Segal-Bargmann space [2,28] F₂( Cⁿ, e^{−| z|2} dg) of functions analytic in z and square-integrable with respect the Gaussian measure e^{−| z|2}dz. Analyticity of \brevef(z) is equivalent to the condition that ( [  ∂/( ∂z_j )] + [ 1/2] z_jI ) Wf(z) equals zero.

The integral in (7.16) is the well-known Segal-Bargmann transform [2,28]. Its inverse is given by a realization of (7.2):

f(x) = ⌠ ⌡ Cn   ^ Wf(z)   w(0,z)(x) dz = ⌠ ⌡ Cn   \brevef(z) e−(z2+x·x)/2+√2zx e− | z|2 dz. (7.17)

This gives (7.2) the name of Segal-Bargmann inverse. The corresponding operator P (7.3) is the identity operator L₂(Rⁿ) → L₂(Rⁿ) and (7.3) gives an integral presentation of the Dirac delta.

Meanwhile the orthoprojection L₂( Cⁿ, e^{− | z|2} dg) →F₂( Cⁿ, e^{− | z|2} dg) is of interest and is a principal ingredient in Berezin quantisation [3,9]. We can easy find its kernel. Indeed, ∧Wf₀(z)=e^{−| z|2}, and the kernel is

K(z,w) = ^ W   f0(z−1·w) - χ   (r(z−1·w)) = ^ W   f0(w−z)exp(ιℑ( - z   ·w) = exp(  1 2 (− | w−z|2 +w· - z   −z· - w   )) = exp(  1 2 (− | z|2− | w|2) +w· - z   ).

To obtain the reproducing kernel for functions \brevef(z)=e^{| z|2} ∧Wf(z) in the Segal-Bargmann space we multiply K(z,w) by e^{(−| z|2+| w|2)/2} which gives the standard reproducing kernel, exp(−| z|² +w·z) [2,(1.10)].

The Segal-Bargmann space is an interesting and important object, but there are also other options. In particular we can consider an alternative representation of the Heisenberg group, this time acting on monogenic functions, an action we introduce in the next subparagraph.

7.3.3  Representation of Hⁿ in Spaces of Monogenic Functions

Let now V_k be the symmetric power monomial defined by the expression

Vk(x)=  1 √ k! (e1 x0−e0 x1)k1 ×(e2 x0− e0 x2)k2 ×…×(en x0− e0 xn)kn.

(7.18)

It can be proved that these monomials are all monogenic (see e.g. [25]), and even that they constitute a basis for the space of monogenic polynomials (as a module over Cl(n)). In general the symmetrized product is not associative, and manipulating it can become quite formal. However, if we restrict the monomials defined above to the hyperplane x₀=0, we obtain

Vk(x)=  1 √ k! x1k1x2k2…xnkn,

and so we have the multiplicative property

 √  k!k′! (k+k)!   VkVk′=Vk+k′,    x0=0.

Another important function is the monogenic exponential function which is defined by

E(u,x)=exp(u·x)   cos(||u|| x0)−  u ||u|| sin(ux0)   .

It is not hard to check [5,§ 14] that this function is monogenic, and of course its restriction to the hyperplane x₀=0 is simply the exponential function, E(u,x)=exp(u·x).

We can therefore extend the symmetric product by the so-called Cauchy-Kovalevskaya product [5,§ 14]: If f and g are monogenic in Rⁿ⁺¹, then f×g is the monogenic function equal to fg on Rⁿ. Introducing the monogenic functions x_i=e_ix₀−e₀x_i we can then write

Vk(x)=  1 √ k! x1k1×x2k2×…×xnkn.

It is fairly easy to check the V_k form an orthonormal set with respect to the following inner product (see [6,§ 3.1] on Clifford valued inner products):

〈 Vk,Vk′ 〉= ⌠ ⌡ Rn+1  - V   k  ( x) Vk′( x) e− | x |2  dx.

(7.19)

Let M₂ be closure of the linear span of {V_k}, using complex coefficients.

The creation and annihilation operators a⁺_k and a⁻_k can be represented by symmetric multiplication (see [25]) with the monogenic variable x_j, which will be written x_k I_×, and by the (classical) partial derivative [ ∂/(∂x_j)]=[ ∂/(∂x_j)] with respect to x_j, which appear in the definition of hypercomplex differentiability. On basis elements they act as follows:

xjI×V(k1,…,kj,…,kn) = √   kj+1   V(k1,…,kj+1,…,kn),  ∂ ∂xj V(k1,…,kj,…,kn) = √   kj   V(k1,…,kj−1,…,kn),

It can be checked that this really is a representation of a^±_k, and that a⁺_k and a⁻_k are each other's adjoint. We use the equalities a⁻_j=[ 1/ √2 ](a⁺_j + a⁻_j) and a⁺_j = [ ι/ √2](a⁻_j − a⁺_j), and the commutation relations [a⁺_i,a⁻_j]=eδ_ij to obtain a representation of the Heisenberg group. Thus an element (t,z), z =u+iv of the Heisenberg group can be written as

(t,z) =   t+  u·u−v·v 4 ,0     0,  (1+ι)(u+v) 2     0,  (1−ι)(u−v) 2   = exp     t+  u2−v2 4   e   exp    (u+v)q √2   exp    (u−v)ιp √2   .

It is therefore represented by the operator

π(t,z) = exp   −   t+  u·u−v·v 4     exp    ((u+v)·x) I× √2   exp    (u−v)·(∂x) √2   , (7.20)

where obviously for a monogenic function f we have

exp    (u−v)ιp √2   f(x) = f   x+  u−v) √2   exp    ((u+v)·x) I× √2   f(x) = E    u+v √2 ,·   ×f(x)

Therefore it is easy to calculate the image of the constant function f₀( x) = V₀(x) ≡ 1, and we obtain the set of functions

f(t,z)(x) = π(t,z) f0(x) = exp   −   t+  u·u−v·v 4     E    u+v √2 ,·   ×f0(x) exp   −   t+  u·u−v·v 4     E    u+v √2 ,x   . (7.21)

In the language of quantum physics f₀(x) is the vacuum vector and functions f_(t,z)(x) are coherent states (or wavelets) for the representation of Hⁿ we described. We can summarize the properties of the representation:

1. All functions in M₂ are complex-vector valued, monogenic in Rⁿ⁺¹, and square integrable with respect to the measure e ^{− | x |2} dx.

2. The representation of the Heisenberg group is given by (7.20). This representation generates a set of coherent states f_(0,z)(x) (7.21) as shifts of the vacuum vector f₀(x) ≡ 1.

3. The creation and annihilation operators a⁺_k and a⁻_k are represented by symmetric (Cauchy-Kovalevskaya) multiplication by x_j and by derivation of monogenic functions. They are adjoint with respect to the inner product (7.19).

4. M₂ is generated as a closed linear space by the orthonormal basis V_k(x)=[ 1/√{k!}](e₁ x₀−e₀ x₁)^k₁ ×(e₂ x₀− e₀ x₂)^k₂ ×…×(e_n x₀− e₀ x_n)^k_n, and also by the set of coherent states f_(t,z)(x) of (7.21).

5. The kernel of the operator intertwining the model constructed here and the Segal-Bargmann one is given by
   

   B(z,x) ∞ ∑ j=0  Vj(x)  zj √ j! =exp( n ∑ k=1  xk - z   k  ),

   which is the holomorphic extension in z =u+ιv of E(u,x). The transformation pair is given by
   

   B f(x) = ⌠ ⌡ Cn  B(z,x)f(z)exp    −|z|2 2    dz B−1φ(z) = ⌠ ⌡ Rn+1  B(z,x)   φ(x)exp    −|x|2 2    dx

6. The space M₂ has a reproducing kernel
   

   K(x,y)= ∞ ∑ k=0  Vk(x) - V   k  (y)= ⌠ ⌡ Cn  B(z,x) B(z,y)    e−| z |2dz.

   Notice that K(x,y) is monogenic in y; it is the monogenic extension of E(y,x).

One can see that some properties of M₂ are closer to those of the Segal-Bargmann space than to those of the space L₂(Rⁿ ) it replaces. It should be noted that the representation of the Heisenberg group we obtained here is new and quite unexpected.

Remark 2 We construct M₂ as a space of complex-vector valued functions. We can also consider an extended space ~M₂ being generated by the orthonormal basis V_k(x) or coherent states f_(0,z)(x) with Clifford valued coefficients multiplied from the right hand side. Such a space will share many properties of M² and have an additional structure: there is a natural representation s: f(x) → s^* f(sxs^*) s of Spin (n) group in ~M₂. Thus this space provides us with a representation of two main symmetries in quantum field theory: the Heisenberg group of quantized coordinate and momentum (external degrees of freedom) and Spin (n) group of quantified inner degrees of freedom. Another composition of the Heisenberg group and Clifford algebras can be found in [19].

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Index (showing section)

Footnotes:

¹ ~G is sometimes called the reduced Heisenberg groupHeisenberg group!reduced. It seems that ~G is a virtual object, which is important in connection with a selected representation of G.